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<span class="kw">let </span><font color="Maroon" title="c1">U</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">U</font><br/>  for <font color="Olive" title="b2">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b3">S</font> being   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a><br/>  for <font color="Olive" title="b4">l1</font>, <font color="Olive" title="b5">l2</font> being   <a href="fomodel1.html#V4" title="FOMODEL1:attr.4">literal</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Olive" title="b3">S</font><br/>  for <font color="Olive" title="b6">I</font> being  <font color="Olive">b<sub>3</sub></font>,<font color="Maroon" title="c1">U</font> <a href="fomodel2.html#V1" title="FOMODEL2:attr.1">-interpreter-like</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Olive" title="b4">l1</font>,<font color="Olive" title="b1">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Olive" title="b6">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Olive" title="b3">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b5">l2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b3">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Olive" title="b5">l2</font>,<font color="Olive" title="b1">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Olive" title="b6">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Olive" title="b5">l2</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Olive" title="b3">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b5">l2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b3">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">U</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b2">S</font> being   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a><br/>  for <font color="Olive" title="b3">l1</font>, <font color="Olive" title="b4">l2</font> being   <a href="fomodel1.html#V4" title="FOMODEL1:attr.4">literal</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Olive" title="b2">S</font><br/>  for <font color="Olive" title="b5">I</font> being  <font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c1">U</font> <a href="fomodel2.html#V1" title="FOMODEL2:attr.1">-interpreter-like</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Olive" title="b3">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Olive" title="b5">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Olive" title="b2">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b4">l2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Olive" title="b4">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Olive" title="b5">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Olive" title="b4">l2</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Olive" title="b2">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b4">l2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a><br/>  for <font color="Olive" title="b2">l1</font>, <font color="Olive" title="b3">l2</font> being   <a href="fomodel1.html#V4" title="FOMODEL1:attr.4">literal</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b4">I</font> being  <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c1">U</font> <a href="fomodel2.html#V1" title="FOMODEL2:attr.1">-interpreter-like</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Olive" title="b2">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Olive" title="b4">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b3">l2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Olive" title="b3">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Olive" title="b4">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Olive" title="b3">l2</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b3">l2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">S</font> be   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">l1</font>, <font color="Olive" title="b2">l2</font> being   <a href="fomodel1.html#V4" title="FOMODEL1:attr.4">literal</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Maroon" title="c4">S</font><br/>  for <font color="Olive" title="b3">I</font> being  <font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">U</font> <a href="fomodel2.html#V1" title="FOMODEL2:attr.1">-interpreter-like</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Olive" title="b1">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Olive" title="b3">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">l2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Olive" title="b2">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Olive" title="b3">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Olive" title="b2">l2</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">l2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">l1</font>, <font color="Maroon" title="c6">l2</font> be   <a href="fomodel1.html#V4" title="FOMODEL1:attr.4">literal</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Maroon" title="c4">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">I</font> being  <font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">U</font> <a href="fomodel2.html#V1" title="FOMODEL2:attr.1">-interpreter-like</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Olive" title="b1">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">l2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Olive" title="b1">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Maroon" title="c6">l2</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">l2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">I</font> be  <font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">U</font> <a href="fomodel2.html#V1" title="FOMODEL2:attr.1">-interpreter-like</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">l2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Maroon" title="c6">l2</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">l2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span><br/>













<span class="kw">set </span><font color="Maroon" title="c8">I1</font> = (<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">I2</font> = (<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">s</font> = <font color="Maroon" title="c5">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Maroon" title="c6">l2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">E1</font> = <span class="p1">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">E2</font> = <span class="p1">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">T</font> = <font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">TT</font> =  <a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c4">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">SS</font> =  <a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c4">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">F</font> = <font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">C</font> = <font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K21" title="FOMODEL1:func.21">-multiCat</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">g1</font> = <font color="Maroon" title="c5">l1</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c2">u</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">g2</font> = <font color="Maroon" title="c6">l2</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c2">u</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">L</font> =  <a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c4">S</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">SSS</font> = <span class="p1">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">l2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c4">S</font></span>)</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c22">D</font> = <font color="Maroon" title="c21">SSS</font> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a> ;<br/>
<a NAME="E2:128"/>
( <font color="Maroon" title="c5">l1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">l1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c6">l2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">l2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E3:128"/><span class="kw">then </span>
( <font color="Maroon" title="c5">l1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c6">l2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">l2</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<a NAME="E4:128"/><span class="kw">then </span>
( <span class="p1">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c5">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">l1</font> &amp; <span class="p1">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c6">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">l2</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">l2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></span>;<br/>
<a NAME="E5:128"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">B2</font></span>: 
( <span class="p1">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c5">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c2">u</font> &amp; <span class="p1">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c6">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c2">u</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T72" title="FUNCOP_1:th.72">FUNCOP_1:72</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span><span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">SSS</font> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Maroon" title="c6">l2</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">SSS</font> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span></span>)</span>;<br/>
<a NAME="E6:128"/><span class="lab"><font color="Green" title="E56">B0</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="128_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c23">T0</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c4">S</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel1.html#T2" title="FOMODEL1:th.2">FOMODEL1:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">T00</font> = <font color="Maroon" title="c23">T0</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c25">ss</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Maroon" title="c6">l2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> as  <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">X0</font> = <font color="Maroon" title="c23">T0</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">SSS</font> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c23">T0</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c27">X</font> = <font color="Maroon" title="c23">T0</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">SSS</font> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c4">S</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E6:128_1"/>
<font color="Maroon" title="c25">ss</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">SSS</font> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span> is <font color="Maroon" title="c23">T0</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c28">sss</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Maroon" title="c6">l2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">T0</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">SSS</font> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span></span>)</span> as  <font color="Maroon" title="c24">T00</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></span>;<br/>
<a NAME="E8:128_1"/><span class="lab"><font color="Green" title="E57">C0</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c28">sss</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">X</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">X</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E9:128_1"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c28">sss</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">U</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">U</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c29">f1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">X</font>, <font color="Maroon" title="c30">f2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c28">sss</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c27">X</font>,<font color="Maroon" title="c1">U</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:128_1"><span class="lab"><font color="Green" title="E57">C0</font></span></a>, <a class="ref" href="funct_2.html#T2" title="FUNCT_2:th.2">FUNCT_2:2</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="128_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c31">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">X</font> implies <font color="Maroon" title="c29">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:128_1_1"/><span class="lab"><font color="Green" title="E58">D0</font></span>: 
<font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c29">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c32">X00</font> = <font color="Maroon" title="c26">X0</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c21">SSS</font> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span> ;<br/><a NAME="E3:128_1_1"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">sss</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">X</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c28">sss</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c23">T0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a>, <a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c33">ssss</font> = <font color="Maroon" title="c28">sss</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c32">X00</font>,<font color="Maroon" title="c23">T0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T2" title="FUNCT_2:th.2">FUNCT_2:2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c34">xx</font> = <font color="Maroon" title="c31">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c32">X00</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_1_1"><span class="lab"><font color="Green" title="E58">D0</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c35">xd</font> = <font color="Maroon" title="c34">xx</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c21">SSS</font> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c36">t0</font> = <font color="Maroon" title="c31">x</font> as   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="fomodel1.html#V14" title="FOMODEL1:attr.14">-termal</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c4">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_1_1"><span class="lab"><font color="Green" title="E58">D0</font></span></a>, <a class="ref" href="fomodel1.html#D33" title="FOMODEL1:def.33">FOMODEL1:def 33</a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c37">t00</font> = <font color="Maroon" title="c36">t0</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c23">T0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel1.html#D33" title="FOMODEL1:def.33">FOMODEL1:def 33</a></span>;<br/><a NAME="E9:128_1_1"/>
<font color="Maroon" title="c23">T0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel1.html#K26" title="FOMODEL1:func.26">AtomicTermsOf</a> <font color="Maroon" title="c4">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel1.html#D30" title="FOMODEL1:def.30">FOMODEL1:def 30</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c38">tl</font> = <font color="Maroon" title="c37">t00</font> as  1 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c4">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T12" title="FOMODEL0:th.12">FOMODEL0:12</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c39">LL</font> = <span class="p1">(<span class="default"><a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">l2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c4">S</font></span>)</span> ;<br/><a NAME="E12:128_1_1"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tl</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c4">S</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c35">xd</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">l2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/><a NAME="E13:128_1_1"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">xx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">l2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/><a NAME="E14:128_1_1"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">xx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="fomodel0.html#K13" title="FOMODEL0:func.13">null</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">l2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T49" title="XBOOLE_1:th.49">XBOOLE_1:49</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c40">xxx</font> = <font color="Maroon" title="c38">tl</font> as  <font color="Maroon" title="c39">LL</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>  1 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/><a NAME="E16:128_1_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c40">xxx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c39">LL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c41">lx</font> = <font color="Maroon" title="c40">xxx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c39">LL</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T60" title="ZFMISC_1:th.60">ZFMISC_1:60</a></span>;<br/><a NAME="E18:128_1_1"/>
 not <font color="Maroon" title="c41">lx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">l2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E19:128_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">Z3</font></span>: 
( <font color="Maroon" title="c41">lx</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">l2</font> &amp;  not <font color="Maroon" title="c41">lx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">l2</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E20:128_1_1"/><span class="lab"><font color="Green" title="E60">Z6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">xxx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c42">newt</font> = (<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c6">l2</font>) <a href="fomodel3.html#K24" title="FOMODEL3:func.24">-SymbolSubstIn</a> <font color="Maroon" title="c36">t0</font> as   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="fomodel1.html#V14" title="FOMODEL1:attr.14">-termal</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c4">S</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c43">s1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c36">t0</font>, <font color="Maroon" title="c44">s2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c42">newt</font> as   <a href="fomodel1.html#V4" title="FOMODEL1:attr.4">literal</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Maroon" title="c4">S</font> ;<br/><a NAME="E23:128_1_1"/><span class="lab"><font color="Green" title="E61">D1</font></span>: 
( <font color="Maroon" title="c43">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">lx</font> &amp; <font color="Maroon" title="c44">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">newt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T6" title="FOMODEL0:th.6">FOMODEL0:6</a></span>;<br/><a NAME="E24:128_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c33">ssss</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">xx</font></span>)</span> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c33">ssss</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c34">xx</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c28">sss</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">xx</font></span>)</span> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Maroon" title="c6">l2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">xx</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c29">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">xx</font></span>)</span> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">xx</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/><a NAME="E25:128_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">D7</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c28">sss</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">sss</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">x</font></span>)</span> &amp; <font color="Maroon" title="c28">sss</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Maroon" title="c6">l2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">x</font> &amp; <font color="Maroon" title="c29">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T29" title="FOMODEL0:th.29">FOMODEL0:29</a></span>;<br/><span class="lab"><font color="Green" title="E63">D4</font></span>: <font color="Maroon" title="c30">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">newt</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:128_1_1"><span class="lab"><font color="Green" title="E62">D7</font></span></a>, <a class="ref" href="fomodel0.html#D23" title="FOMODEL0:def.23">FOMODEL0:def 23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c44">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K33" title="FOMODEL1:func.33">SubTerms</a> <font color="Maroon" title="c42">newt</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel2.html#T21" title="FOMODEL2:th.21">FOMODEL2:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c44">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>

;<br/><span class="lab"><font color="Green" title="E64">D5</font></span>: <font color="Maroon" title="c29">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c43">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K33" title="FOMODEL1:func.33">SubTerms</a> <font color="Maroon" title="c36">t0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:128_1_1"><span class="lab"><font color="Green" title="E62">D7</font></span></a>, <a class="ref" href="fomodel2.html#T21" title="FOMODEL2:th.21">FOMODEL2:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c43">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>

;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_1_1_3"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:128_1_1_3"/>
( <font color="Maroon" title="c41">lx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l1</font> or <font color="Maroon" title="c41">lx</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">l1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_1_1_3_1"><span class="kw">suppose </span></a><a NAME="E1:128_1_1_3_1"/><span class="lab"><font color="Green" title="E65">E0</font></span>: 
<font color="Maroon" title="c41">lx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c29">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:128_1_1_3_1"/>
<font color="Maroon" title="c36">t0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><font color="Maroon" title="c5">l1</font></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:128_1_1"><span class="lab"><font color="Green" title="E60">Z6</font></span></a>, <a class="txt" href="#E1:128_1_1_3_1"><span class="lab"><font color="Green" title="E65">E0</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/><a NAME="E3:128_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c42">newt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><font color="Maroon" title="c6">l2</font></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T34" title="FOMODEL0:th.34">FOMODEL0:34</a></span>;<br/><span class="kw">hence </span><a NAME="E4:128_1_1_3_1"/>
<font color="Maroon" title="c29">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_1_1_3_1"><span class="lab"><font color="Green" title="E65">E0</font></span></a>, <a class="txt" href="#E27:128_1_1"><span class="lab"><font color="Green" title="E64">D5</font></span></a>, <a class="txt" href="#E5:128"><span class="lab"><font color="Green" title="E55">B2</font></span></a>, <a class="txt" href="#E26:128_1_1"><span class="lab"><font color="Green" title="E63">D4</font></span></a>, <a class="txt" href="#E23:128_1_1"><span class="lab"><font color="Green" title="E61">D1</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_1_1_3_2"><span class="kw">suppose </span></a><a NAME="E1:128_1_1_3_2"/>
<font color="Maroon" title="c41">lx</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">l1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c29">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:128_1_1_3_2"/><span class="kw">then </span>
( <font color="Maroon" title="c42">newt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">lx</font> &amp;  not <font color="Maroon" title="c41">lx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">l1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T105" title="FUNCT_4:th.105">FUNCT_4:105</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:128_1_1_3_2"/><span class="kw">then </span>
( <font color="Maroon" title="c44">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">lx</font> &amp;  not <font color="Maroon" title="c41">lx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T6" title="FOMODEL0:th.6">FOMODEL0:6</a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><a NAME="E4:128_1_1_3_2"/><span class="kw">then </span>
( <font color="Maroon" title="c30">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">lx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> &amp; <font color="Maroon" title="c29">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">lx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:128_1_1"><span class="lab"><font color="Green" title="E61">D1</font></span></a>, <a class="txt" href="#E26:128_1_1"><span class="lab"><font color="Green" title="E63">D4</font></span></a>, <a class="txt" href="#E27:128_1_1"><span class="lab"><font color="Green" title="E64">D5</font></span></a>, <a class="txt" href="#E19:128_1_1"><span class="lab"><font color="Green" title="E59">Z3</font></span></a>, <a class="ref" href="funct_4.html#T11" title="FUNCT_4:th.11">FUNCT_4:11</a></span>;<br/><span class="kw">hence </span><a NAME="E5:128_1_1_3_2"/>
<font color="Maroon" title="c29">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E12:128_1"/><span class="kw">then </span>
<font color="Maroon" title="c29">f1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c30">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:128_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T83" title="RELAT_1:th.83">RELAT_1:83</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:128"/><span class="lab"><font color="Green" title="E57">B1</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="128_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c23">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c24">nn</font> = <font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c25">NN</font> = <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:128_2"/><span class="lab"><font color="Green" title="E58">C1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c23">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c26">Tnn</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c24">nn</font>, <font color="Maroon" title="c27">TNN</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c25">NN</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c4">S</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel1.html#T2" title="FOMODEL1:th.2">FOMODEL1:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c28">Xn</font> = <font color="Maroon" title="c26">Tnn</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">SSS</font> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span>, <font color="Maroon" title="c29">X</font> = <font color="Maroon" title="c27">TNN</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">SSS</font> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c4">S</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E5:128_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Maroon" title="c6">l2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c25">NN</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">SSS</font> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span> is <font color="Maroon" title="c27">TNN</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c30">sX</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Maroon" title="c6">l2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c29">X</font> as   <a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c4">S</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></span>;<br/>
<a NAME="E7:128_2"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Maroon" title="c6">l2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c29">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">X</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c30">sX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c4">S</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a>, <a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c31">sXX</font> = <font color="Maroon" title="c30">sX</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c29">X</font>,<span class="p1">(<span class="default"><a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c4">S</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T2" title="FUNCT_2:th.2">FUNCT_2:2</a></span>;<br/>
<a NAME="E9:128_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Maroon" title="c6">l2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c24">nn</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">SSS</font> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span> is <font color="Maroon" title="c26">Tnn</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c32">sXn</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Maroon" title="c6">l2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c28">Xn</font> as   <a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c4">S</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></span>;<br/>
<a NAME="E11:128_2"/><span class="lab"><font color="Green" title="E59">C0</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c29">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">X</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c30">sX</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">X</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c32">sXn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">Xn</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E12:128_2"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c29">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">U</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c30">sX</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">U</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c33">f1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c29">X</font>, <font color="Maroon" title="c34">f2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c30">sX</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c29">X</font>,<font color="Maroon" title="c1">U</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:128_2"><span class="lab"><font color="Green" title="E59">C0</font></span></a>, <a class="ref" href="funct_2.html#T2" title="FUNCT_2:th.2">FUNCT_2:2</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="128_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c35">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">X</font> implies <font color="Maroon" title="c34">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:128_2_1"/><span class="lab"><font color="Green" title="E60">D0</font></span>: 
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c34">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c36">XX</font> = <font color="Maroon" title="c29">X</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c27">TNN</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c37">xxx</font> = <font color="Maroon" title="c35">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c36">XX</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_2_1"><span class="lab"><font color="Green" title="E60">D0</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c38">xx</font> = <font color="Maroon" title="c37">xxx</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c21">SSS</font> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_2_1"><span class="lab"><font color="Green" title="E60">D0</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c39">tN</font> = <font color="Maroon" title="c35">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c27">TNN</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_2_1"><span class="lab"><font color="Green" title="E60">D0</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c40">t</font> = <font color="Maroon" title="c39">tN</font> as  <font color="Maroon" title="c25">NN</font> <a href="fomodel1.html#V14" title="FOMODEL1:attr.14">-termal</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c4">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel1.html#D33" title="FOMODEL1:def.33">FOMODEL1:def 33</a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c41">ttt</font> = <font color="Maroon" title="c40">t</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c4">S</font> ;<br/><span class="kw">set </span><font color="Maroon" title="c42">d</font> =  <a href="fomodel1.html#K38" title="FOMODEL1:func.38">Depth</a> <font color="Maroon" title="c40">t</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c43">aux</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">xx</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">l2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/><a NAME="E9:128_2_1"/>
( <font color="Maroon" title="c43">aux</font> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">SSS</font> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K33" title="FOMODEL1:func.33">SubTerms</a> <font color="Maroon" title="c40">t</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c40">t</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/><a NAME="E10:128_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">D0</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K33" title="FOMODEL1:func.33">SubTerms</a> <font color="Maroon" title="c40">t</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">SSS</font> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c44">tt</font> = <font color="Maroon" title="c40">t</font> as  <font color="Maroon" title="c24">nn</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="fomodel1.html#V14" title="FOMODEL1:attr.14">-termal</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c4">S</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c45">newt</font> = (<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c6">l2</font>) <a href="fomodel3.html#K24" title="FOMODEL3:func.24">-SymbolSubstIn</a> <font color="Maroon" title="c40">t</font> as  <font color="Maroon" title="c25">NN</font> <a href="fomodel1.html#V14" title="FOMODEL1:attr.14">-termal</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c4">S</font> ;<br/><a NAME="E13:128_2_1"/><span class="lab"><font color="Green" title="E62">Z1</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K33" title="FOMODEL1:func.33">SubTerms</a> <font color="Maroon" title="c44">tt</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c24">nn</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/><a NAME="E14:128_2_1"/><span class="lab"><font color="Green" title="E63">D2</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c28">Xn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">Xn</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Maroon" title="c6">l2</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c28">Xn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">Xn</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:128_2"><span class="lab"><font color="Green" title="E59">C0</font></span></a>, <a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a>, <a class="ref" href="relat_1.html#T83" title="RELAT_1:th.83">RELAT_1:83</a></span>;<br/><a NAME="E15:128_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c33">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">xxx</font></span>)</span> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">xxx</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Maroon" title="c6">l2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c36">XX</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">xxx</font></span>)</span> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Maroon" title="c6">l2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">xxx</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c31">sXX</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">xxx</font></span>)</span> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">sXX</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">xxx</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/><a NAME="E16:128_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">D3</font></span>: 
( <font color="Maroon" title="c33">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">x</font> &amp; <font color="Maroon" title="c30">sX</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Maroon" title="c6">l2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">x</font> &amp; <font color="Maroon" title="c34">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">sX</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">x</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T29" title="FOMODEL0:th.29">FOMODEL0:29</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:128_2_1_1"/>
(  <a href="fomodel1.html#K38" title="FOMODEL1:func.38">Depth</a> <font color="Maroon" title="c40">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="fomodel1.html#K38" title="FOMODEL1:func.38">Depth</a> <font color="Maroon" title="c40">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:128_2_1_1_1"/>
 <a href="fomodel1.html#K38" title="FOMODEL1:func.38">Depth</a> <font color="Maroon" title="c40">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c34">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c46">t0</font> = <font color="Maroon" title="c40">t</font> as   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="fomodel1.html#V14" title="FOMODEL1:attr.14">-termal</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c4">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel1.html#D40" title="FOMODEL1:def.40">FOMODEL1:def 40</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c47">l</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c46">t0</font> as   <a href="fomodel1.html#V4" title="FOMODEL1:attr.4">literal</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Maroon" title="c4">S</font> ;<br/><span class="lab"><font color="Green" title="E65">E0</font></span>: <font color="Maroon" title="c46">t0</font> = 
<span class="p1"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><font color="Maroon" title="c47">l</font></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K21" title="FOMODEL1:func.21">-multiCat</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K33" title="FOMODEL1:func.33">SubTerms</a> <font color="Maroon" title="c46">t0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel1.html#D37" title="FOMODEL1:def.37">FOMODEL1:def 37</a></span>
<br/>.= 
<span class="p1"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><font color="Maroon" title="c47">l</font></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel3.html#K25" title="FOMODEL3:func.25">null</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>

;<br/><a NAME="E5:128_2_1_1_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c38">xx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c21">SSS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E6:128_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c46">t0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">SSS</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T60" title="ZFMISC_1:th.60">ZFMISC_1:60</a></span>;<br/><a NAME="E7:128_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c47">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">SSS</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T6" title="FOMODEL0:th.6">FOMODEL0:6</a></span>;<br/><a NAME="E8:128_2_1_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c47">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">l2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E9:128_2_1_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c47">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">l2</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><a NAME="E10:128_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">E2</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c47">l</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">I</font> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c47">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T11" title="FUNCT_4:th.11">FUNCT_4:11</a></span>;<br/><span class="lab"><font color="Green" title="E67">E1</font></span>: <font color="Maroon" title="c33">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">t0</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c47">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K33" title="FOMODEL1:func.33">SubTerms</a> <font color="Maroon" title="c46">t0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:128_2_1"><span class="lab"><font color="Green" title="E64">D3</font></span></a>, <a class="ref" href="fomodel2.html#T21" title="FOMODEL2:th.21">FOMODEL2:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c47">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>

;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_2_1_1_1_3"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:128_2_1_1_1_3"/>
( <font color="Maroon" title="c47">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span> or  not <font color="Maroon" title="c47">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_2_1_1_1_3_1"><span class="kw">suppose </span></a><a NAME="E1:128_2_1_1_1_3_1"/>
<font color="Maroon" title="c47">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c34">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:128_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">ZZ0</font></span>: 
<font color="Maroon" title="c47">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">l1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><a NAME="E3:128_2_1_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c47">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E4:128_2_1_1_1_3_1"/><span class="kw">then </span>
( <font color="Maroon" title="c33">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">t0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c5">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Maroon" title="c6">l2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c46">t0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><font color="Maroon" title="c6">l2</font></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:128_2_1_1_1"><span class="lab"><font color="Green" title="E67">E1</font></span></a>, <a class="txt" href="#E4:128_2_1_1_1"><span class="lab"><font color="Green" title="E65">E0</font></span></a>, <a class="ref" href="fomodel0.html#T35" title="FOMODEL0:th.35">FOMODEL0:35</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c34">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">t0</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><font color="Maroon" title="c6">l2</font></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K33" title="FOMODEL1:func.33">SubTerms</a> <span class="p3"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><font color="Maroon" title="c6">l2</font></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:128_2_1"><span class="lab"><font color="Green" title="E64">D3</font></span></a>, <a class="ref" href="fomodel2.html#T21" title="FOMODEL2:th.21">FOMODEL2:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><font color="Maroon" title="c6">l2</font></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T6" title="FOMODEL0:th.6">FOMODEL0:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c6">l2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:128_2_1_1_1_3_1"/>
<font color="Maroon" title="c34">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:128_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E68">ZZ0</font></span></a>, <a class="txt" href="#E11:128_2_1_1_1"><span class="lab"><font color="Green" title="E67">E1</font></span></a>, <a class="txt" href="#E5:128"><span class="lab"><font color="Green" title="E55">B2</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_2_1_1_1_3_2"><span class="kw">suppose </span></a><a NAME="E1:128_2_1_1_1_3_2"/><span class="lab"><font color="Green" title="E68">Z0</font></span>: 
 not <font color="Maroon" title="c47">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c34">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:128_2_1_1_1_3_2"/><span class="kw">then </span>
( <font color="Maroon" title="c33">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">t0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c47">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> &amp;  not <font color="Maroon" title="c47">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">l1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:128_2_1_1_1"><span class="lab"><font color="Green" title="E67">E1</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a>, <a class="ref" href="funct_4.html#T11" title="FUNCT_4:th.11">FUNCT_4:11</a></span>;<br/><a NAME="E3:128_2_1_1_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c47">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">l1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E4:128_2_1_1_1_3_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Maroon" title="c6">l2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c46">t0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c46">t0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:128_2_1_1_1"><span class="lab"><font color="Green" title="E65">E0</font></span></a>, <a class="ref" href="fomodel0.html#T35" title="FOMODEL0:th.35">FOMODEL0:35</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c34">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">t0</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c47">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K33" title="FOMODEL1:func.33">SubTerms</a> <font color="Maroon" title="c46">t0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:128_2_1"><span class="lab"><font color="Green" title="E64">D3</font></span></a>, <a class="ref" href="fomodel2.html#T21" title="FOMODEL2:th.21">FOMODEL2:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c47">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>

;<br/><span class="kw">hence </span><a NAME="E6:128_2_1_1_1_3_2"/>
<font color="Maroon" title="c34">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128_2_1_1_1_3_2"><span class="lab"><font color="Green" title="E68">Z0</font></span></a>, <a class="txt" href="#E11:128_2_1_1_1"><span class="lab"><font color="Green" title="E67">E1</font></span></a>, <a class="txt" href="#E10:128_2_1_1_1"><span class="lab"><font color="Green" title="E66">E2</font></span></a>, <a class="ref" href="funct_4.html#T11" title="FUNCT_4:th.11">FUNCT_4:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:128_2_1_1_2"/>
 <a href="fomodel1.html#K38" title="FOMODEL1:func.38">Depth</a> <font color="Maroon" title="c40">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c34">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:128_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">E0</font></span>: 
 not <font color="Maroon" title="c40">t</font> is  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="fomodel1.html#V14" title="FOMODEL1:attr.14">-termal</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c46">s1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c40">t</font> as   non  <a href="fomodel1.html#V4" title="FOMODEL1:attr.4">literal</a>   <a href="fomodel1.html#V10" title="FOMODEL1:attr.10">ofAtomicFormula</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Maroon" title="c4">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel1.html#T16" title="FOMODEL1:th.16">FOMODEL1:16</a></span>;<br/><a NAME="E4:128_2_1_1_2"/>
( <font color="Maroon" title="c40">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c40">t</font> &amp;  not <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c40">t</font> is  <a href="fomodel1.html#V4" title="FOMODEL1:attr.4">literal</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:128_2_1_1_2"><span class="lab"><font color="Green" title="E65">E0</font></span></a>, <a class="ref" href="fomodel0.html#T6" title="FOMODEL0:th.6">FOMODEL0:6</a>, <a class="ref" href="fomodel1.html#T16" title="FOMODEL1:th.16">FOMODEL1:16</a></span>;<br/><a NAME="E5:128_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c45">newt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c46">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T105" title="FUNCT_4:th.105">FUNCT_4:105</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c47">s2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c45">newt</font> as   non  <a href="fomodel1.html#V4" title="FOMODEL1:attr.4">literal</a>   <a href="fomodel1.html#V10" title="FOMODEL1:attr.10">ofAtomicFormula</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Maroon" title="c4">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T6" title="FOMODEL0:th.6">FOMODEL0:6</a></span>;<br/><a NAME="E7:128_2_1_1_2"/>
(  not <font color="Maroon" title="c46">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">l1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  not <font color="Maroon" title="c47">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">l2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E8:128_2_1_1_2"/><span class="kw">then </span>
(  not <font color="Maroon" title="c46">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span> &amp;  not <font color="Maroon" title="c47">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">l2</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><a NAME="E9:128_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">E1</font></span>: 
( <span class="p1">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">s1</font> &amp; <span class="p1">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">s2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T11" title="FUNCT_4:th.11">FUNCT_4:11</a></span>;<br/><a NAME="E10:128_2_1_1_2"/>
<font color="Maroon" title="c46">s1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">l1</font>
 ;<br/><a NAME="E11:128_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c40">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">l1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T6" title="FOMODEL0:th.6">FOMODEL0:6</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c45">newt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<font color="Maroon" title="c40">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T105" title="FUNCT_4:th.105">FUNCT_4:105</a></span>
<br/>.= 
<font color="Maroon" title="c46">s1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T6" title="FOMODEL0:th.6">FOMODEL0:6</a></span>
;<br/><a NAME="E13:128_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">E2</font></span>: 
<font color="Maroon" title="c46">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T6" title="FOMODEL0:th.6">FOMODEL0:6</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c34">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">newt</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:128_2_1"><span class="lab"><font color="Green" title="E64">D3</font></span></a>, <a class="ref" href="fomodel0.html#D23" title="FOMODEL0:def.23">FOMODEL0:def 23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K33" title="FOMODEL1:func.33">SubTerms</a> <font color="Maroon" title="c45">newt</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel2.html#T21" title="FOMODEL2:th.21">FOMODEL2:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Maroon" title="c6">l2</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K33" title="FOMODEL1:func.33">SubTerms</a> <font color="Maroon" title="c40">t</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel3.html#E65"><span class="lab"><font color="Green" title="E52">Lm40</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Maroon" title="c6">l2</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K33" title="FOMODEL1:func.33">SubTerms</a> <font color="Maroon" title="c40">t</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:128_2_1_1_2"><span class="lab"><font color="Green" title="E66">E1</font></span></a>, <a class="ref" href="relat_1.html#T36" title="RELAT_1:th.36">RELAT_1:36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Maroon" title="c6">l2</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">SSS</font> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K33" title="FOMODEL1:func.33">SubTerms</a> <font color="Maroon" title="c40">t</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:128_2_1"><span class="lab"><font color="Green" title="E62">Z1</font></span></a>, <a class="txt" href="#E10:128_2_1"><span class="lab"><font color="Green" title="E61">D0</font></span></a>, <a class="txt" href="#E14:128_2_1"><span class="lab"><font color="Green" title="E63">D2</font></span></a>, <a class="ref" href="relat_1.html#T165" title="RELAT_1:th.165">RELAT_1:165</a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K33" title="FOMODEL1:func.33">SubTerms</a> <font color="Maroon" title="c40">t</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:128_2_1"><span class="lab"><font color="Green" title="E62">Z1</font></span></a>, <a class="txt" href="#E10:128_2_1"><span class="lab"><font color="Green" title="E61">D0</font></span></a>, <a class="txt" href="#E14:128_2_1"><span class="lab"><font color="Green" title="E63">D2</font></span></a>, <a class="txt" href="#E2:128_2"><span class="lab"><font color="Green" title="E58">C1</font></span></a>, <a class="ref" href="relat_1.html#T165" title="RELAT_1:th.165">RELAT_1:165</a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>
<br/>.= 
<font color="Maroon" title="c33">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:128_2_1"><span class="lab"><font color="Green" title="E64">D3</font></span></a>, <a class="txt" href="#E9:128_2_1_1_2"><span class="lab"><font color="Green" title="E66">E1</font></span></a>, <a class="txt" href="#E13:128_2_1_1_2"><span class="lab"><font color="Green" title="E67">E2</font></span></a>, <a class="ref" href="fomodel2.html#T21" title="FOMODEL2:th.21">FOMODEL2:21</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E15:128_2"/><span class="kw">then </span>
<font color="Maroon" title="c33">f1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c34">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:128_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T83" title="RELAT_1:th.83">RELAT_1:83</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:128"/>
 for <font color="Olive" title="b1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E6:128"><span class="lab"><font color="Green" title="E56">B0</font></span></a>, <a class="txt" href="#E7:128"><span class="lab"><font color="Green" title="E57">B1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E9:128"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c5">l1</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">l2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c6">l2</font>,<font color="Maroon" title="c2">u</font>) <a href="fomodel2.html#K5" title="FOMODEL2:func.5">ReassignIn</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l1</font> <a href="fomodel0.html#K20" title="FOMODEL0:func.20">SubstWith</a> <font color="Maroon" title="c6">l2</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">l2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
