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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="2" unmarked=""><NAME>and</NAME></SYMBOL><SYMBOL arity="2" unmarked="a__plus"><NAME>Marked_a__plus</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>a__plus</NAME></SYMBOL><SYMBOL arity="1" unmarked="mark"><NAME>Marked_mark</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>s</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>x</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>a__and</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>a__x</NAME></SYMBOL><SYMBOL arity="2" unmarked="a__x"><NAME>Marked_a__x</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>tt</NAME></SYMBOL><SYMBOL arity="2" unmarked="a__and"><NAME>Marked_a__and</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>0</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>plus</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>mark</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>X2</VAR><VAR>X1</VAR><VAR>M</VAR><VAR>N</VAR><VAR>X</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>a__and(tt,X)</LHS><RHS>mark(X)</RHS></RULE><RULE><LHS>a__plus(N,0)</LHS><RHS>mark(N)</RHS></RULE><RULE><LHS>a__plus(N,s(M))</LHS><RHS>s(a__plus(mark(N),mark(M)))</RHS></RULE><RULE><LHS>a__x(N,0)</LHS><RHS>0</RHS></RULE><RULE><LHS>a__x(N,s(M))</LHS><RHS>a__plus(a__x(mark(N),mark(M)),mark(N))</RHS></RULE><RULE><LHS>mark(and(X1,X2))</LHS><RHS>a__and(mark(X1),X2)</RHS></RULE><RULE><LHS>mark(plus(X1,X2))</LHS><RHS>a__plus(mark(X1),mark(X2))</RHS></RULE><RULE><LHS>mark(x(X1,X2))</LHS><RHS>a__x(mark(X1),mark(X2))</RHS></RULE><RULE><LHS>mark(tt)</LHS><RHS>tt</RHS></RULE><RULE><LHS>mark(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>mark(s(X))</LHS><RHS>s(mark(X))</RHS></RULE><RULE><LHS>a__and(X1,X2)</LHS><RHS>and(X1,X2)</RHS></RULE><RULE><LHS>a__plus(X1,X2)</LHS><RHS>plus(X1,X2)</RHS></RULE><RULE><LHS>a__x(X1,X2)</LHS><RHS>x(X1,X2)</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>a__and(tt,X)</LHS><RHS>mark(X)</RHS></RULE><RULE><LHS>a__plus(N,0)</LHS><RHS>mark(N)</RHS></RULE><RULE><LHS>a__plus(N,s(M))</LHS><RHS>s(a__plus(mark(N),mark(M)))</RHS></RULE><RULE><LHS>a__x(N,0)</LHS><RHS>0</RHS></RULE><RULE><LHS>a__x(N,s(M))</LHS><RHS>a__plus(a__x(mark(N),mark(M)),mark(N))</RHS></RULE><RULE><LHS>mark(and(X1,X2))</LHS><RHS>a__and(mark(X1),X2)</RHS></RULE><RULE><LHS>mark(plus(X1,X2))</LHS><RHS>a__plus(mark(X1),mark(X2))</RHS></RULE><RULE><LHS>mark(x(X1,X2))</LHS><RHS>a__x(mark(X1),mark(X2))</RHS></RULE><RULE><LHS>mark(tt)</LHS><RHS>tt</RHS></RULE><RULE><LHS>mark(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>mark(s(X))</LHS><RHS>s(mark(X))</RHS></RULE><RULE><LHS>a__and(X1,X2)</LHS><RHS>and(X1,X2)</RHS></RULE><RULE><LHS>a__plus(X1,X2)</LHS><RHS>plus(X1,X2)</RHS></RULE><RULE><LHS>a__x(X1,X2)</LHS><RHS>x(X1,X2)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_a__and(tt,X)</LHS><RHS>Marked_mark(X)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_a__plus(N,0)</LHS><RHS>Marked_mark(N)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_a__plus(N,s(M))</LHS><RHS>Marked_a__plus(mark(N),mark(M))</RHS></DPRULE><DPRULE num="3"><LHS>Marked_a__plus(N,s(M))</LHS><RHS>Marked_mark(N)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_a__plus(N,s(M))</LHS><RHS>Marked_mark(M)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_a__x(N,s(M))</LHS><RHS>Marked_a__plus(a__x(mark(N),mark(M)),mark(N))</RHS></DPRULE><DPRULE num="6"><LHS>Marked_a__x(N,s(M))</LHS><RHS>Marked_a__x(mark(N),mark(M))</RHS></DPRULE><DPRULE num="7"><LHS>Marked_a__x(N,s(M))</LHS><RHS>Marked_mark(N)</RHS></DPRULE><DPRULE num="8"><LHS>Marked_a__x(N,s(M))</LHS><RHS>Marked_mark(M)</RHS></DPRULE><DPRULE num="9"><LHS>Marked_a__x(N,s(M))</LHS><RHS>Marked_mark(N)</RHS></DPRULE><DPRULE num="10"><LHS>Marked_mark(and(X1,X2))</LHS><RHS>Marked_a__and(mark(X1),X2)</RHS></DPRULE><DPRULE num="11"><LHS>Marked_mark(and(X1,X2))</LHS><RHS>Marked_mark(X1)</RHS></DPRULE><DPRULE num="12"><LHS>Marked_mark(plus(X1,X2))</LHS><RHS>Marked_a__plus(mark(X1),mark(X2))</RHS></DPRULE><DPRULE num="13"><LHS>Marked_mark(plus(X1,X2))</LHS><RHS>Marked_mark(X1)</RHS></DPRULE><DPRULE num="14"><LHS>Marked_mark(plus(X1,X2))</LHS><RHS>Marked_mark(X2)</RHS></DPRULE><DPRULE num="15"><LHS>Marked_mark(x(X1,X2))</LHS><RHS>Marked_a__x(mark(X1),mark(X2))</RHS></DPRULE><DPRULE num="16"><LHS>Marked_mark(x(X1,X2))</LHS><RHS>Marked_mark(X1)</RHS></DPRULE><DPRULE num="17"><LHS>Marked_mark(x(X1,X2))</LHS><RHS>Marked_mark(X2)</RHS></DPRULE><DPRULE num="18"><LHS>Marked_mark(s(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="0"/><NODE ref="1"/><NODE ref="2"/><NODE ref="3"/><NODE ref="4"/><NODE ref="5"/><NODE ref="6"/><NODE ref="7"/><NODE ref="8"/><NODE ref="7"/><NODE ref="10"/><NODE ref="11"/><NODE ref="12"/><NODE ref="13"/><NODE ref="14"/><NODE ref="15"/><NODE ref="16"/><NODE ref="17"/><NODE ref="18"/></SCC></CCLIST></DAG></CRITERION><PROPERTY criterion="stronggraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>a__and(tt,X)</LHS><RHS>mark(X)</RHS></RULE><RULE><LHS>a__plus(N,0)</LHS><RHS>mark(N)</RHS></RULE><RULE><LHS>a__plus(N,s(M))</LHS><RHS>s(a__plus(mark(N),mark(M)))</RHS></RULE><RULE><LHS>a__x(N,0)</LHS><RHS>0</RHS></RULE><RULE><LHS>a__x(N,s(M))</LHS><RHS>a__plus(a__x(mark(N),mark(M)),mark(N))</RHS></RULE><RULE><LHS>mark(and(X1,X2))</LHS><RHS>a__and(mark(X1),X2)</RHS></RULE><RULE><LHS>mark(plus(X1,X2))</LHS><RHS>a__plus(mark(X1),mark(X2))</RHS></RULE><RULE><LHS>mark(x(X1,X2))</LHS><RHS>a__x(mark(X1),mark(X2))</RHS></RULE><RULE><LHS>mark(tt)</LHS><RHS>tt</RHS></RULE><RULE><LHS>mark(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>mark(s(X))</LHS><RHS>s(mark(X))</RHS></RULE><RULE><LHS>a__and(X1,X2)</LHS><RHS>and(X1,X2)</RHS></RULE><RULE><LHS>a__plus(X1,X2)</LHS><RHS>plus(X1,X2)</RHS></RULE><RULE><LHS>a__x(X1,X2)</LHS><RHS>x(X1,X2)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_a__and(tt,X)</LHS><RHS>Marked_mark(X)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_a__plus(N,0)</LHS><RHS>Marked_mark(N)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_a__plus(N,s(M))</LHS><RHS>Marked_a__plus(mark(N),mark(M))</RHS></DPRULE><DPRULE num="3"><LHS>Marked_a__plus(N,s(M))</LHS><RHS>Marked_mark(N)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_a__plus(N,s(M))</LHS><RHS>Marked_mark(M)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_a__x(N,s(M))</LHS><RHS>Marked_a__plus(a__x(mark(N),mark(M)),mark(N))</RHS></DPRULE><DPRULE num="6"><LHS>Marked_a__x(N,s(M))</LHS><RHS>Marked_a__x(mark(N),mark(M))</RHS></DPRULE><DPRULE num="7"><LHS>Marked_a__x(N,s(M))</LHS><RHS>Marked_mark(N)</RHS></DPRULE><DPRULE num="8"><LHS>Marked_a__x(N,s(M))</LHS><RHS>Marked_mark(M)</RHS></DPRULE><DPRULE num="9"><LHS>Marked_a__x(N,s(M))</LHS><RHS>Marked_mark(N)</RHS></DPRULE><DPRULE num="10"><LHS>Marked_mark(and(X1,X2))</LHS><RHS>Marked_a__and(mark(X1),X2)</RHS></DPRULE><DPRULE num="11"><LHS>Marked_mark(and(X1,X2))</LHS><RHS>Marked_mark(X1)</RHS></DPRULE><DPRULE num="12"><LHS>Marked_mark(plus(X1,X2))</LHS><RHS>Marked_a__plus(mark(X1),mark(X2))</RHS></DPRULE><DPRULE num="13"><LHS>Marked_mark(plus(X1,X2))</LHS><RHS>Marked_mark(X1)</RHS></DPRULE><DPRULE num="14"><LHS>Marked_mark(plus(X1,X2))</LHS><RHS>Marked_mark(X2)</RHS></DPRULE><DPRULE num="15"><LHS>Marked_mark(x(X1,X2))</LHS><RHS>Marked_a__x(mark(X1),mark(X2))</RHS></DPRULE><DPRULE num="16"><LHS>Marked_mark(x(X1,X2))</LHS><RHS>Marked_mark(X1)</RHS></DPRULE><DPRULE num="17"><LHS>Marked_mark(x(X1,X2))</LHS><RHS>Marked_mark(X2)</RHS></DPRULE><DPRULE num="18"><LHS>Marked_mark(s(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="rpo"><AFS><PROJARG num="0"><SYMBOL><NAME>Marked_mark</NAME></SYMBOL></PROJARG><PROJARG num="0"><SYMBOL><NAME>mark</NAME></SYMBOL></PROJARG></AFS><STATUSLIST><STATUS status="lex"><SYMBOL><NAME>plus</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>Marked_a__and</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>Marked_a__x</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>a__x</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>a__and</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>x</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>s</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>a__plus</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>Marked_a__plus</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>and</NAME></SYMBOL></STATUS></STATUSLIST><PRECEDENCE><PREC><OLHS><SYMBOL><NAME>a__plus</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>a__x</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>a__plus</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>x</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>a__plus</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>Marked_a__x</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>s</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>a__plus</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>s</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>a__x</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>s</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>plus</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>s</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>x</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>s</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>Marked_a__x</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>plus</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>a__x</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>plus</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>x</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>plus</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>Marked_a__x</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>Marked_a__plus</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>a__plus</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>Marked_a__plus</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>a__x</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>Marked_a__plus</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>plus</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>Marked_a__plus</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>x</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>Marked_a__plus</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>Marked_a__x</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>Marked_a__and</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>a__and</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>Marked_a__and</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>and</NAME></SYMBOL></ORHS></PREC></PRECEDENCE><EQUIVALENCE><EQ><OLHS><SYMBOL><NAME>a__and</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>and</NAME></SYMBOL></ORHS></EQ><EQ><OLHS><SYMBOL><NAME>a__plus</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>plus</NAME></SYMBOL></ORHS></EQ><EQ><OLHS><SYMBOL><NAME>a__x</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>x</NAME></SYMBOL></ORHS></EQ><EQ><OLHS><SYMBOL><NAME>a__x</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>Marked_a__x</NAME></SYMBOL></ORHS></EQ><EQ><OLHS><SYMBOL><NAME>and</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>a__and</NAME></SYMBOL></ORHS></EQ><EQ><OLHS><SYMBOL><NAME>plus</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>a__plus</NAME></SYMBOL></ORHS></EQ><EQ><OLHS><SYMBOL><NAME>x</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>a__x</NAME></SYMBOL></ORHS></EQ><EQ><OLHS><SYMBOL><NAME>x</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>Marked_a__x</NAME></SYMBOL></ORHS></EQ><EQ><OLHS><SYMBOL><NAME>Marked_a__x</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>a__x</NAME></SYMBOL></ORHS></EQ><EQ><OLHS><SYMBOL><NAME>Marked_a__x</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>x</NAME></SYMBOL></ORHS></EQ></EQUIVALENCE></ORDERING><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_a__and(tt,X)</LHS><RHS>Marked_mark(X)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_a__plus(N,0)</LHS><RHS>Marked_mark(N)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_a__plus(N,s(M))</LHS><RHS>Marked_a__plus(mark(N),mark(M))</RHS></DPRULE><DPRULE num="3"><LHS>Marked_a__plus(N,s(M))</LHS><RHS>Marked_mark(N)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_a__plus(N,s(M))</LHS><RHS>Marked_mark(M)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_a__x(N,s(M))</LHS><RHS>Marked_a__plus(a__x(mark(N),mark(M)),mark(N))</RHS></DPRULE><DPRULE num="6"><LHS>Marked_a__x(N,s(M))</LHS><RHS>Marked_a__x(mark(N),mark(M))</RHS></DPRULE><DPRULE num="7"><LHS>Marked_a__x(N,s(M))</LHS><RHS>Marked_mark(N)</RHS></DPRULE><DPRULE num="8"><LHS>Marked_a__x(N,s(M))</LHS><RHS>Marked_mark(M)</RHS></DPRULE><DPRULE num="9"><LHS>Marked_a__x(N,s(M))</LHS><RHS>Marked_mark(N)</RHS></DPRULE><DPRULE num="10"><LHS>Marked_mark(and(X1,X2))</LHS><RHS>Marked_a__and(mark(X1),X2)</RHS></DPRULE><DPRULE num="11"><LHS>Marked_mark(and(X1,X2))</LHS><RHS>Marked_mark(X1)</RHS></DPRULE><DPRULE num="12"><LHS>Marked_mark(plus(X1,X2))</LHS><RHS>Marked_a__plus(mark(X1),mark(X2))</RHS></DPRULE><DPRULE num="13"><LHS>Marked_mark(plus(X1,X2))</LHS><RHS>Marked_mark(X1)</RHS></DPRULE><DPRULE num="14"><LHS>Marked_mark(plus(X1,X2))</LHS><RHS>Marked_mark(X2)</RHS></DPRULE><DPRULE num="15"><LHS>Marked_mark(x(X1,X2))</LHS><RHS>Marked_mark(X1)</RHS></DPRULE><DPRULE num="16"><LHS>Marked_mark(x(X1,X2))</LHS><RHS>Marked_mark(X2)</RHS></DPRULE><DPRULE num="17"><LHS>Marked_mark(s(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE></DPLIST></STRICTPAIRS></CRITERION><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>a__and(tt,X)</LHS><RHS>mark(X)</RHS></RULE><RULE><LHS>a__plus(N,0)</LHS><RHS>mark(N)</RHS></RULE><RULE><LHS>a__plus(N,s(M))</LHS><RHS>s(a__plus(mark(N),mark(M)))</RHS></RULE><RULE><LHS>a__x(N,0)</LHS><RHS>0</RHS></RULE><RULE><LHS>a__x(N,s(M))</LHS><RHS>a__plus(a__x(mark(N),mark(M)),mark(N))</RHS></RULE><RULE><LHS>mark(and(X1,X2))</LHS><RHS>a__and(mark(X1),X2)</RHS></RULE><RULE><LHS>mark(plus(X1,X2))</LHS><RHS>a__plus(mark(X1),mark(X2))</RHS></RULE><RULE><LHS>mark(x(X1,X2))</LHS><RHS>a__x(mark(X1),mark(X2))</RHS></RULE><RULE><LHS>mark(tt)</LHS><RHS>tt</RHS></RULE><RULE><LHS>mark(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>mark(s(X))</LHS><RHS>s(mark(X))</RHS></RULE><RULE><LHS>a__and(X1,X2)</LHS><RHS>and(X1,X2)</RHS></RULE><RULE><LHS>a__plus(X1,X2)</LHS><RHS>plus(X1,X2)</RHS></RULE><RULE><LHS>a__x(X1,X2)</LHS><RHS>x(X1,X2)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_mark(x(X1,X2))</LHS><RHS>Marked_a__x(mark(X1),mark(X2))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><NONSCC num="0"><NODE ref="0"/></NONSCC></CCLIST></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
