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<span class="kw">let </span><font color="Maroon" title="c1">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">  for <font color="Olive" title="b1">U</font> being   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> <br/>  for <font color="Olive" title="b2">S1</font>, <font color="Olive" title="b3">S2</font> being   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a><br/>  for <font color="Olive" title="b4">I1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Olive" title="b2">S1</font><br/>  for <font color="Olive" title="b5">I2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Olive" title="b3">S2</font>  st <font color="Olive" title="b4">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> &amp;  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Olive" title="b2">S1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Olive" title="b3">S2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b4">I1</font> <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"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">I2</font> <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"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">S1</font>, <font color="Olive" title="b2">S2</font> being   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a><br/>  for <font color="Olive" title="b3">I1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Olive" title="b1">S1</font><br/>  for <font color="Olive" title="b4">I2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Olive" title="b2">S2</font>  st <font color="Olive" title="b3">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> &amp;  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Olive" title="b1">S1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Olive" title="b2">S2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">I1</font> <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"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">I2</font> <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"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">S1</font>, <font color="Maroon" title="c4">S2</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">I1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c3">S1</font><br/>  for <font color="Olive" title="b2">I2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c4">S2</font>  st <font color="Olive" title="b1">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> &amp;  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c3">S1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c4">S2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">I1</font> <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"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">I2</font> <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"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span></span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">T1</font> = <font color="Maroon" title="c3">S1</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">O1</font> =  <a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c3">S1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">TT1</font> =  <a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c3">S1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">SS1</font> =  <a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c3">S1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">L1</font> =  <a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c3">S1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">F1</font> = <font color="Maroon" title="c3">S1</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">C1</font> = <font color="Maroon" title="c3">S1</font> <a href="fomodel1.html#K21" title="FOMODEL1:func.21">-multiCat</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">AT1</font> =  <a href="fomodel1.html#K26" title="FOMODEL1:func.26">AtomicTermsOf</a> <font color="Maroon" title="c3">S1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">II1</font> = <font color="Maroon" title="c2">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c3">S1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">a1</font> =  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c3">S1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">strings1</font> = <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="c3">S1</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">T2</font> = <font color="Maroon" title="c4">S2</font> <a href="fomodel1.html#K39" title="FOMODEL1:func.39">-termsOfMaxDepth</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">O2</font> =  <a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c4">S2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">TT2</font> =  <a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c4">S2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">SS2</font> =  <a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c4">S2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">L2</font> =  <a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c4">S2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">F2</font> = <font color="Maroon" title="c4">S2</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c22">C2</font> = <font color="Maroon" title="c4">S2</font> <a href="fomodel1.html#K21" title="FOMODEL1:func.21">-multiCat</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c23">AT2</font> =  <a href="fomodel1.html#K26" title="FOMODEL1:func.26">AtomicTermsOf</a> <font color="Maroon" title="c4">S2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c24">II2</font> = <font color="Maroon" title="c2">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c4">S2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c25">a2</font> =  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c4">S2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c26">strings2</font> = <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">S2</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c27">I1</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c3">S1</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">I2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c4">S2</font>  st <font color="Maroon" title="c27">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> &amp;  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c3">S1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c4">S2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c27">I1</font> <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"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">I2</font> <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"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c28">I2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c4">S2</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="c27">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> &amp;  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c3">S1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c4">S2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c27">I1</font> <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"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <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"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> )</span><br/>



<a NAME="E1:152"/><span class="lab"><font color="Green" title="E68">B3</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c27">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c3">S1</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c28">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c4">S2</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/>
<span class="kw">set </span><font color="Maroon" title="c29">E1</font> = <font color="Maroon" title="c27">I1</font> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c30">E2</font> = <font color="Maroon" title="c28">I2</font> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c31">I11</font> = <font color="Maroon" title="c27">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c32">I22</font> = <font color="Maroon" title="c28">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">assume </span><a NAME="E2:152"/><span class="lab"><font color="Green" title="E69">Z2</font></span>: 
<font color="Maroon" title="c27">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c3">S1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c4">S2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c27">I1</font> <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"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <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"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> )</span><br/>

<a NAME="E3:152"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">B2</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">I1</font> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c3">S1</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c4">S2</font> &amp; <font color="Maroon" title="c27">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E71">BB4</font></span>: <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c27">I1</font></span>)</span> = 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:152"><span class="lab"><font color="Green" title="E69">Z2</font></span></a>, <a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>
<br/>.= 
<font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c28">I2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</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"><font color="Maroon" title="c27">I1</font> <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"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</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="c3">S1</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></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <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"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</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">S2</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></span>)</span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1"> { <span class="default"> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of $1 : verum </span> } </span> ;<br/>
<a NAME="E5:152"/><span class="lab"><font color="Green" title="E72">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="152_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:152_2"/><span class="lab"><font color="Green" title="E73">Z3</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">S2</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="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">S2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">S1</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="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c3">S1</font> )
 
<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="c33">D1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></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"><font color="Maroon" title="c3">S1</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   <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="c3">S1</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:152_2"><span class="lab"><font color="Green" title="E73">Z3</font></span></a>, <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="c34">D2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></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"><font color="Maroon" title="c4">S2</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   <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">S2</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:152_2"><span class="lab"><font color="Green" title="E73">Z3</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c35">f1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c33">D1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c36">f2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c34">D2</font>;<br/>
<a NAME="E4:152_2"/><span class="lab"><font color="Green" title="E74">C1</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"><font color="Maroon" title="c27">I1</font> <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="c33">D1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">D1</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"><font color="Maroon" title="c28">I2</font> <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="c34">D2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">D2</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="152_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c37">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="c37">x</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"><span class="p2">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c33">D1</font></span>)</span> implies ( <font color="Maroon" title="c37">x</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"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c34">D2</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c34">D2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">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="c27">I1</font> <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="c33">D1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:152_2_1"/><span class="lab"><font color="Green" title="E75">Z7</font></span>: 
<font color="Maroon" title="c37">x</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"><span class="p2">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c33">D1</font></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="c37">x</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"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c34">D2</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c34">D2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">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="c27">I1</font> <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="c33">D1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font> )</span><br/><a NAME="E2:152_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">C7</font></span>: 
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K26" title="FOMODEL1:func.26">AtomicTermsOf</a> <font color="Maroon" title="c3">S1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:152_2"><span class="lab"><font color="Green" title="E74">C1</font></span></a>, <a class="ref" href="fomodel1.html#D30" title="FOMODEL1:def.30">FOMODEL1:def 30</a></span>;<br/><a NAME="E3:152_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">C6</font></span>: 
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c3">S1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T2" title="FOMODEL0:th.2">FOMODEL0:2</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c38">Y1</font> = <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c3">S1</font></span>)</span> as   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>  ;<br/><a NAME="E5:152_2_1"/>
<font color="Maroon" title="c38">Y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="c39">XX</font> = <font color="Maroon" title="c1">X</font> as   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>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c40">xx</font> = <font color="Maroon" title="c37">x</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:152_2_1"><span class="lab"><font color="Green" title="E75">Z7</font></span></a>, <a class="txt" href="#E4:152_2"><span class="lab"><font color="Green" title="E74">C1</font></span></a></span>;<br/><a NAME="E8:152_2_1"/>
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c38">Y1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:152_2_1"><span class="lab"><font color="Green" title="E77">C6</font></span></a>, <a class="ref" href="finseq_2.html#T96" title="FINSEQ_2:th.96">FINSEQ_2:96</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c41">y1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c38">Y1</font><span class="kw"> such that </span><br/><a NAME="E10:152_2_1"/><span class="lab"><font color="Green" title="E78">C0</font></span>: 
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c41">y1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ;<br/><a NAME="E11:152_2_1"/>
<font color="Maroon" title="c41">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c38">Y1</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c42">l1</font> = <font color="Maroon" title="c41">y1</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="c3">S1</font> ;<br/><a NAME="E13:152_2_1"/>
( <font color="Maroon" title="c42">l1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c42">l1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c3">S1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel1.html#D19" title="FOMODEL1:def.19">FOMODEL1:def 19</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E14:152_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c42">l1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c4">S2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:152"><span class="lab"><font color="Green" title="E71">BB4</font></span></a>, <a class="txt" href="#E1:152"><span class="lab"><font color="Green" title="E68">B3</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c43">s2</font> = <font color="Maroon" title="c42">l1</font> as   <a href="fomodel1.html#V9" title="FOMODEL1:attr.9">own</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Maroon" title="c4">S2</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c44">s22</font> = <font color="Maroon" title="c43">s2</font>, <font color="Maroon" title="c45">l11</font> = <font color="Maroon" title="c42">l1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c39">XX</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E17:152_2_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c39">XX</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">s22</font></span>)</span> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">s22</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"><font color="Maroon" title="c27">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c39">XX</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">l11</font></span>)</span> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">l11</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="E18:152_2_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">I2</font> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c43">s2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c27">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">I1</font> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c42">l1</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="kw">then </span><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c2">U</font> = 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c43">s2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:152"><span class="lab"><font color="Green" title="E69">Z2</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <font color="Maroon" title="c43">s2</font></span>)</span></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c2">U</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
;<br/><a NAME="E20:152_2_1"/><span class="kw">then </span>
(  not <font color="Maroon" title="c43">s2</font> is  <a href="fomodel1.html#V7" title="FOMODEL1:attr.7">relational</a>  &amp;  not <font color="Maroon" title="c43">s2</font> is  <a href="fomodel1.html#V6" title="FOMODEL1:attr.6">operational</a>  )
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c46">l2</font> = <font color="Maroon" title="c43">s2</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">S2</font> ;<br/><a NAME="E22:152_2_1"/>
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel1.html#K26" title="FOMODEL1:func.26">AtomicTermsOf</a> <font color="Maroon" title="c3">S1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:152_2_1"><span class="lab"><font color="Green" title="E76">C7</font></span></a></span>;<br/><a NAME="E23:152_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">S1</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 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="c47">t1</font> = <font color="Maroon" title="c37">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="c3">S1</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/><span class="kw">reconsider </span><font color="Maroon" title="c48">D11</font> = <font color="Maroon" title="c33">D1</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="c3">S1</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:152_2_1"><span class="lab"><font color="Green" title="E75">Z7</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c49">x1</font> = <font color="Maroon" title="c47">t1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c48">D11</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:152_2_1"><span class="lab"><font color="Green" title="E75">Z7</font></span></a>, <a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/><a NAME="E27:152_2_1"/><span class="lab"><font color="Green" title="E79">CC8</font></span>: 
( <font color="Maroon" title="c45">l11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c39">XX</font> &amp; <font color="Maroon" title="c46">l2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c4">S2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel1.html#D14" title="FOMODEL1:def.14">FOMODEL1:def 14</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c50">Y2</font> = <font color="Maroon" title="c39">XX</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c4">S2</font></span>)</span> as   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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c51">ll2</font> = <font color="Maroon" title="c46">l2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c50">Y2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:152_2_1"><span class="lab"><font color="Green" title="E79">CC8</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E30:152_2_1"/>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c51">ll2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c50">Y2</font>)
 ;<br/><a NAME="E31:152_2_1"/><span class="kw">then </span>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c51">ll2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c50">Y2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T96" title="FINSEQ_2:th.96">FINSEQ_2:96</a></span>;<br/><a NAME="E32:152_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">CC11</font></span>: 
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K26" title="FOMODEL1:func.26">AtomicTermsOf</a> <font color="Maroon" title="c4">S2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:152_2_1"><span class="lab"><font color="Green" title="E78">C0</font></span></a>, <a class="ref" href="fomodel0.html#T2" title="FOMODEL0:th.2">FOMODEL0:2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c52">D22</font> = <font color="Maroon" title="c34">D2</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">S2</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:152_2_1"><span class="lab"><font color="Green" title="E80">CC11</font></span></a>, <a class="ref" href="fomodel1.html#D30" title="FOMODEL1:def.30">FOMODEL1:def 30</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c53">x2</font> = <font color="Maroon" title="c37">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c52">D22</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:152_2_1"><span class="lab"><font color="Green" title="E80">CC11</font></span></a>, <a class="ref" href="fomodel1.html#D30" title="FOMODEL1:def.30">FOMODEL1:def 30</a></span>;<br/><a NAME="E35:152_2_1"/><span class="lab"><font color="Green" title="E81">C10</font></span>: 
<font color="Maroon" title="c47">t1</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">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:152_2_1"><span class="lab"><font color="Green" title="E78">C0</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c54">y11</font> = <font color="Maroon" title="c47">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c39">XX</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E37:152_2_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c39">XX</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c54">y11</font></span>)</span> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c54">y11</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"><font color="Maroon" title="c28">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c39">XX</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c54">y11</font></span>)</span> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c54">y11</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="c28">I2</font> <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="c52">D22</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c53">x2</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="c28">I2</font> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c53">x2</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="E38:152_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">C12</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c27">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c54">y11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c54">y11</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c54">y11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">I2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c54">y11</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c53">x2</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="c28">I2</font> <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="c52">D22</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c53">x2</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/><a NAME="E39:152_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c48">D11</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">x1</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="c27">I1</font> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c49">x1</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/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c33">D1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c47">t1</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="p1">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c3">S1</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="c47">t1</font></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"><font color="Maroon" title="c27">I1</font> <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="c47">t1</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"><font color="Maroon" title="c28">I2</font> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c46">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="txt" href="#E35:152_2_1"><span class="lab"><font color="Green" title="E81">C10</font></span></a>, <a class="txt" href="#E2:152"><span class="lab"><font color="Green" title="E69">Z2</font></span></a>, <a class="txt" href="#E38:152_2_1"><span class="lab"><font color="Green" title="E82">C12</font></span></a>, <a class="ref" href="fomodel0.html#T6" title="FOMODEL0:th.6">FOMODEL0:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c46">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="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <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">S2</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="c46">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"><font color="Maroon" title="c28">I2</font> <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="c46">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="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="c28">I2</font> <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="c34">D2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:152_2_1"><span class="lab"><font color="Green" title="E78">C0</font></span></a>, <a class="txt" href="#E38:152_2_1"><span class="lab"><font color="Green" title="E82">C12</font></span></a>, <a class="ref" href="fomodel2.html#T21" title="FOMODEL2:th.21">FOMODEL2:21</a></span>
;<br/><span class="kw">hence </span><a NAME="E41:152_2_1"/>
( <font color="Maroon" title="c37">x</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"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c34">D2</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c34">D2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">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="c27">I1</font> <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="c33">D1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:152_2"><span class="lab"><font color="Green" title="E74">C1</font></span></a>, <a class="txt" href="#E32:152_2_1"><span class="lab"><font color="Green" title="E80">CC11</font></span></a>, <a class="ref" href="fomodel1.html#D30" title="FOMODEL1:def.30">FOMODEL1:def 30</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>
<span class="kw">hence </span><a NAME="E6:152_2"/>
<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="fomodel0.html#T51" title="FOMODEL0:th.51">FOMODEL0:51</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>
<span class="kw">assume </span><a NAME="E6:152"/><span class="lab"><font color="Green" title="E73">B5</font></span>: 
 the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c3">S1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c4">S2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font>
 ; <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"><font color="Maroon" title="c27">I1</font> <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"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <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"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span></span><br/>

<a NAME="E7:152"/><span class="lab"><font color="Green" title="E74">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="152_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c33">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="c33">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c33">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:152_3"/><span class="lab"><font color="Green" title="E75">C0</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c33">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="c33">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="c34">nn</font> = <font color="Maroon" title="c33">n</font>, <font color="Maroon" title="c35">NN</font> = <font color="Maroon" title="c33">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/>
<a NAME="E3:152_3"/><span class="lab"><font color="Green" title="E76">Z3</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">S2</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="c35">NN</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">S2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">S1</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="c35">NN</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="c3">S1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">S1</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="c34">nn</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="c3">S1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">S2</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="c34">nn</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">S2</font> )
 
<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="c36">D1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></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"><font color="Maroon" title="c3">S1</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="c35">NN</font></span>)</span>, <font color="Maroon" title="c37">d1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></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"><font color="Maroon" title="c3">S1</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="c34">nn</font></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="c3">S1</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:152_3"><span class="lab"><font color="Green" title="E76">Z3</font></span></a>, <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="c38">D2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></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"><font color="Maroon" title="c4">S2</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="c35">NN</font></span>)</span>, <font color="Maroon" title="c39">d2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></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"><font color="Maroon" title="c4">S2</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="c34">nn</font></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">S2</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:152_3"><span class="lab"><font color="Green" title="E76">Z3</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c40">f1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c36">D1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c41">f2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c38">D2</font>;<br/>
<a NAME="E6:152_3"/><span class="lab"><font color="Green" title="E77">C1</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"><font color="Maroon" title="c27">I1</font> <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="c36">D1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">D1</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"><font color="Maroon" title="c28">I2</font> <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="c38">D2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">D2</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"><font color="Maroon" title="c27">I1</font> <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="c37">d1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">d1</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"><font color="Maroon" title="c28">I2</font> <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="c39">d2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">d2</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="E7:152_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">C2</font></span>: 
<font color="Maroon" title="c37">d1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c39">d2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:152_3"><span class="lab"><font color="Green" title="E75">C0</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c42">d12</font> = <font color="Maroon" title="c37">d1</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c39">d2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:152_3"><span class="lab"><font color="Green" title="E75">C0</font></span></a>, <a class="txt" href="#E6:152_3"><span class="lab"><font color="Green" title="E77">C1</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c43">d21</font> = <font color="Maroon" title="c42">d12</font> 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">S2</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="152_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c44">y</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="c44">y</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"><span class="p2">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c36">D1</font></span>)</span> implies ( <font color="Olive">b<sub>1</sub></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"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c38">D2</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c36">D1</font></span>)</span> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c38">D2</font></span>)</span> <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="E1:152_3_1"/><span class="lab"><font color="Green" title="E79">D20</font></span>: 
<font color="Maroon" title="c44">y</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"><span class="p2">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c36">D1</font></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="Olive">b<sub>1</sub></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"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c38">D2</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c36">D1</font></span>)</span> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c38">D2</font></span>)</span> <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="c45">D11</font> = <font color="Maroon" title="c36">D1</font> as   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>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c46">y1</font> = <font color="Maroon" title="c44">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c45">D11</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:152_3_1"><span class="lab"><font color="Green" title="E79">D20</font></span></a>, <a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/><a NAME="E4:152_3_1"/>
<font color="Maroon" title="c46">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">S1</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="c35">NN</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c47">t1</font> = <font color="Maroon" title="c46">y1</font> as  <font color="Maroon" title="c34">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="c3">S1</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/><span class="kw">reconsider </span><font color="Maroon" title="c48">o1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">S1</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="c47">t1</font> as   <a href="fomodel1.html#V8" title="FOMODEL1:attr.8">termal</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Maroon" title="c3">S1</font> ;<br/><span class="kw">set </span><font color="Maroon" title="c49">m1</font> =  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <font color="Maroon" title="c48">o1</font></span>)</span>;<br/><a NAME="E7:152_3_1"/><span class="lab"><font color="Green" title="E80">D9</font></span>: 
( <font color="Maroon" title="c47">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp;  not <font color="Maroon" title="c47">t1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c50">XX</font> = <font color="Maroon" title="c1">X</font> as   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>  ;<br/><a NAME="E9:152_3_1"/><span class="lab"><font color="Green" title="E81">DD1</font></span>: 
<font color="Maroon" title="c46">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c50">XX</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E10:152_3_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="c47">t1</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="c50">XX</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="#E9:152_3_1"><span class="lab"><font color="Green" title="E81">DD1</font></span></a></span>;<br/><a NAME="E11:152_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">DD3</font></span>: 
( <font color="Maroon" title="c47">t1</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="c50">XX</font> &amp; <font color="Maroon" title="c48">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">t1</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>, <a class="ref" href="zfmisc_1.html#T60" title="ZFMISC_1:th.60">ZFMISC_1:60</a></span>;<br/><a NAME="E12:152_3_1"/><span class="kw">then </span>
( <font color="Maroon" title="c48">o1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c50">XX</font> &amp; <font color="Maroon" title="c48">o1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c3">S1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel1.html#D19" title="FOMODEL1:def.19">FOMODEL1:def 19</a></span>;<br/><a NAME="E13:152_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c48">o1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c50">XX</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c4">S2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:152"><span class="lab"><font color="Green" title="E71">BB4</font></span></a>, <a class="txt" href="#E1:152"><span class="lab"><font color="Green" title="E68">B3</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c51">o2</font> = <font color="Maroon" title="c48">o1</font> as   <a href="fomodel1.html#V9" title="FOMODEL1:attr.9">own</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Maroon" title="c4">S2</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c52">o22</font> = <font color="Maroon" title="c51">o2</font> as   <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">S2</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c53">ox</font> = <font color="Maroon" title="c48">o1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c50">XX</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:152_3_1"><span class="lab"><font color="Green" title="E82">DD3</font></span></a></span>;<br/><a NAME="E17:152_3_1"/>
( <span class="p1">(<span class="default"> the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c3">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c53">ox</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"> the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c3">S1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c50">XX</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c53">ox</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"> the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c53">ox</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"> the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c4">S2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c50">XX</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c53">ox</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="E18:152_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">D19</font></span>: 
(  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c3">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c48">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c3">S1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c48">o1</font> &amp;  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c51">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c4">S2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c51">o2</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/><a NAME="E19:152_3_1"/>
 <a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <font color="Maroon" title="c48">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <font color="Maroon" title="c52">o22</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:152_3_1"><span class="lab"><font color="Green" title="E83">D19</font></span></a>, <a class="txt" href="#E6:152"><span class="lab"><font color="Green" title="E73">B5</font></span></a></span>;<br/><a NAME="E20:152_3_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c52">o22</font> is  <a href="fomodel1.html#V7" title="FOMODEL1:attr.7">relational</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c54">o2</font> = <font color="Maroon" title="c51">o2</font> as   <a href="fomodel1.html#V8" title="FOMODEL1:attr.8">termal</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Maroon" title="c4">S2</font> ;<br/><span class="kw">set </span><font color="Maroon" title="c55">m2</font> =  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <font color="Maroon" title="c54">o2</font></span>)</span>;<br/><a NAME="E22:152_3_1"/><span class="lab"><font color="Green" title="E84">D14</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c27">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c53">ox</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="c27">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c50">XX</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c53">ox</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="c28">I2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c53">ox</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="c28">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c50">XX</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c53">ox</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/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">D17</font></span>: <font color="Maroon" title="c27">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c53">ox</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c50">XX</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c48">o1</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/>.= 
<font color="Maroon" title="c28">I2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c53">ox</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:152_3_1"><span class="lab"><font color="Green" title="E84">D14</font></span></a>, <a class="txt" href="#E2:152"><span class="lab"><font color="Green" title="E69">Z2</font></span></a>, <a class="ref" href="fomodel0.html#T29" title="FOMODEL0:th.29">FOMODEL0:29</a></span>
;<br/><span class="kw">set </span><font color="Maroon" title="c56">st1</font> =  <a href="fomodel1.html#K33" title="FOMODEL1:func.33">SubTerms</a> <font color="Maroon" title="c47">t1</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c57">B</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c47">t1</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="c50">XX</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:152_3_1"><span class="lab"><font color="Green" title="E81">DD1</font></span></a>, <a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/><a NAME="E25:152_3_1"/>
(  <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="c47">t1</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="c47">t1</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp; <font color="Maroon" title="c57">B</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="c50">XX</font> <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="E26:152_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">D2</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="c47">t1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c50">XX</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</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/><a NAME="E27:152_3_1"/><span class="lab"><font color="Green" title="E87">DD10</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="c47">t1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">S1</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="c33">n</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="E28:152_3_1"/><span class="kw">then </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="c47">t1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c37">d1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:152_3_1"><span class="lab"><font color="Green" title="E86">D2</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/><a NAME="E29:152_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">D7</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="c47">t1</font></span>)</span> <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#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c3">S1</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &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="c47">t1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c39">d2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:152_3"><span class="lab"><font color="Green" title="E78">C2</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E30:152_3_1"/><span class="kw">then </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="c47">t1</font></span>)</span> <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#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c3">S1</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &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="c47">t1</font></span>)</span> <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#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c4">S2</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="E31:152_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">DD6</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="c47">t1</font></span>)</span> <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="c3">S1</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</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="c47">t1</font></span>)</span> <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">S2</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="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E32:152_3_1"/>
 <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="c47">t1</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">S2</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="c34">nn</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:152_3_1"><span class="lab"><font color="Green" title="E88">D7</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E33:152_3_1"/><span class="kw">then </span>
 <a href="fomodel1.html#K33" title="FOMODEL1:func.33">SubTerms</a> <font color="Maroon" title="c47">t1</font> is <span class="p1">(<span class="default"><font color="Maroon" title="c4">S2</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="c34">nn</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</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="E34:152_3_1"/><span class="kw">then </span>
 <a href="fomodel1.html#K33" title="FOMODEL1:func.33">SubTerms</a> <font color="Maroon" title="c47">t1</font> is   <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <font color="Maroon" title="c54">o2</font></span>)</span> <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 <span class="p1">(<span class="default"><font color="Maroon" title="c4">S2</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="c34">nn</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:152"><span class="lab"><font color="Green" title="E73">B5</font></span></a>, <a class="txt" href="#E18:152_3_1"><span class="lab"><font color="Green" title="E83">D19</font></span></a>, <a class="ref" href="fomodel0.html#T26" title="FOMODEL0:th.26">FOMODEL0:26</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c58">st2</font> =  <a href="fomodel1.html#K33" title="FOMODEL1:func.33">SubTerms</a> <font color="Maroon" title="c47">t1</font> as   <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <font color="Maroon" title="c54">o2</font></span>)</span> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">S2</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="c34">nn</font></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c59">T2n</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">S2</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="c34">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">S2</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/><a NAME="E37:152_3_1"/>
<font color="Maroon" title="c58">st2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c59">T2n</font> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c60">st22</font> = <font color="Maroon" title="c58">st2</font> as   <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <font color="Maroon" title="c54">o2</font></span>)</span> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c4">S2</font></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c61">t2</font> = <font color="Maroon" title="c54">o2</font> <a href="fomodel3.html#K1" title="FOMODEL3:func.1">-compound</a> <font color="Maroon" title="c58">st2</font> as  <font color="Maroon" title="c34">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">S2</font> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="152_3_1_2"><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="E1:152_3_1_2"/>
( <font color="Maroon" title="c47">t1</font> is  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="fomodel1.html#V14" title="FOMODEL1:attr.14">-termal</a>  or  not <font color="Maroon" title="c47">t1</font> is  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="fomodel1.html#V14" title="FOMODEL1:attr.14">-termal</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="152_3_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:152_3_1_2_1"/>
<font color="Maroon" title="c47">t1</font> is  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="fomodel1.html#V14" title="FOMODEL1:attr.14">-termal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>1</sub></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"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c38">D2</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c36">D1</font></span>)</span> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c38">D2</font></span>)</span> <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="c62">t11</font> = <font color="Maroon" title="c47">t1</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="c3">S1</font> ;<br/><a NAME="E3:152_3_1_2_1"/><span class="lab"><font color="Green" title="E90">EE2</font></span>: 
( <font color="Maroon" title="c62">t11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp; <font color="Maroon" title="c62">t11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">S1</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 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/><a NAME="E4:152_3_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">E2</font></span>: 
<font color="Maroon" title="c62">t11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c50">XX</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></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"><font color="Maroon" title="c3">S1</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:152_3_1_2_1"/><span class="lab"><font color="Green" title="E92">EE3</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">S2</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="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">S2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">S1</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="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c3">S1</font> )
 <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="c63">A1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></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"><font color="Maroon" title="c3">S1</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   <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="c3">S1</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:152_3_1_2_1"><span class="lab"><font color="Green" title="E92">EE3</font></span></a>, <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="c64">A2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></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"><font color="Maroon" title="c4">S2</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   <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">S2</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:152_3_1_2_1"><span class="lab"><font color="Green" title="E92">EE3</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c65">g1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c63">A1</font>;<br/><span class="kw">set </span><font color="Maroon" title="c66">g2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c64">A2</font>;<br/><a NAME="E8:152_3_1_2_1"/><span class="lab"><font color="Green" title="E93">E1</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"><font color="Maroon" title="c27">I1</font> <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="c63">A1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c63">A1</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"><font color="Maroon" title="c28">I2</font> <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="c64">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c64">A2</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="E9:152_3_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">EE4</font></span>: 
<font color="Maroon" title="c63">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c64">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:152"><span class="lab"><font color="Green" title="E72">B0</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>;<br/><a NAME="E10:152_3_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c62">t11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c64">A2</font> &amp; <font color="Maroon" title="c62">t11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c63">A1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:152_3_1_2_1"><span class="lab"><font color="Green" title="E91">E2</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c67">t2</font> = <font color="Maroon" title="c62">t11</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">S2</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="E12:152_3_1_2_1"/>
<font color="Maroon" title="c67">t2</font> is <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c35">NN</font></span>)</span> <a href="fomodel1.html#V14" title="FOMODEL1:attr.14">-termal</a> 
 ;<br/><a NAME="E13:152_3_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c67">t2</font> is <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">NN</font> <a href="fomodel1.html#V14" title="FOMODEL1:attr.14">-termal</a> 
 ;<br/><a NAME="E14:152_3_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">EE6</font></span>: 
( <font color="Maroon" title="c67">t2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c50">XX</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp; <font color="Maroon" title="c67">t2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S2</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="c35">NN</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">thus </span><a NAME="E15:152_3_1_2_1"/>
<font color="Maroon" title="c44">y</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"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c38">D2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:152_3"><span class="lab"><font color="Green" title="E77">C1</font></span></a>, <a class="txt" href="#E14:152_3_1_2_1"><span class="lab"><font color="Green" title="E95">EE6</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <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="c27">I1</font> <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="c36">D1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">y</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="c28">I2</font> <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="c38">D2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">y</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c68">D22</font> = <font color="Maroon" title="c38">D2</font> as   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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:152_3_1_2_1"><span class="lab"><font color="Green" title="E95">EE6</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c69">A11</font> = <font color="Maroon" title="c63">A1</font>, <font color="Maroon" title="c70">A22</font> = <font color="Maroon" title="c64">A2</font> as   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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:152_3_1_2_1"><span class="lab"><font color="Green" title="E93">E1</font></span></a>, <a class="txt" href="#E5:152"><span class="lab"><font color="Green" title="E72">B0</font></span></a>, <a class="txt" href="#E3:152_3_1_2_1"><span class="lab"><font color="Green" title="E90">EE2</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c71">t111</font> = <font color="Maroon" title="c62">t11</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c69">A11</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:152_3_1_2_1"><span class="lab"><font color="Green" title="E90">EE2</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c72">t02</font> = <font color="Maroon" title="c62">t11</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c70">A22</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:152_3_1_2_1"><span class="lab"><font color="Green" title="E94">EE4</font></span></a>, <a class="txt" href="#E4:152_3_1_2_1"><span class="lab"><font color="Green" title="E91">E2</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c73">t20</font> = <font color="Maroon" title="c67">t2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c68">D22</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:152_3_1_2_1"><span class="lab"><font color="Green" title="E95">EE6</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E21:152_3_1_2_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c45">D11</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">y1</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="c27">I1</font> <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="c46">y1</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="c27">I1</font> <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="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">S1</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></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c71">t111</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="c27">I1</font> <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="c71">t111</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="c28">I2</font> <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="c64">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c72">t02</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="c28">I2</font> <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="c72">t02</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="c28">I2</font> <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="c68">D22</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c73">t20</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="c28">I2</font> <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="c73">t20</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="E22:152_3_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">E5</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c36">D1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c44">y</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">S1</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></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c44">y</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">S2</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></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c44">y</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c38">D2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c44">y</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="kw">thus </span><a NAME="E23:152_3_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c36">D1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">y</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="c28">I2</font> <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="c38">D2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:152_3_1_2_1"><span class="lab"><font color="Green" title="E96">E5</font></span></a>, <a class="txt" href="#E5:152"><span class="lab"><font color="Green" title="E72">B0</font></span></a>, <a class="txt" href="#E8:152_3_1_2_1"><span class="lab"><font color="Green" title="E93">E1</font></span></a>, <a class="txt" href="#E4:152_3_1_2_1"><span class="lab"><font color="Green" title="E91">E2</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</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="152_3_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:152_3_1_2_2"/>
 not <font color="Maroon" title="c47">t1</font> is  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="fomodel1.html#V14" title="FOMODEL1:attr.14">-termal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>1</sub></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"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c38">D2</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c36">D1</font></span>)</span> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c38">D2</font></span>)</span> <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="E2:152_3_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c48">o1</font> is  <a href="fomodel1.html#V6" title="FOMODEL1:attr.6">operational</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel1.html#T16" title="FOMODEL1:th.16">FOMODEL1:16</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c62">n1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:152_3_1_2_2"/><span class="lab"><font color="Green" title="E90">E0</font></span>: 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <font color="Maroon" title="c48">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c62">n1</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="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c63">nn1</font> = <font color="Maroon" title="c62">n1</font> 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">reconsider </span><font color="Maroon" title="c64">st11</font> =  <a href="fomodel1.html#K33" title="FOMODEL1:func.33">SubTerms</a> <font color="Maroon" title="c47">t1</font> as  <font color="Maroon" title="c63">nn1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c3">S1</font></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:152_3_1_2_2"><span class="lab"><font color="Green" title="E90">E0</font></span></a></span>;<br/><a NAME="E7:152_3_1_2_2"/><span class="lab"><font color="Green" title="E91">E1</font></span>: 
 not <font color="Maroon" title="c64">st11</font> is <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a> 
 ;<br/><a NAME="E8:152_3_1_2_2"/>
<font color="Maroon" title="c64">st11</font> is <span class="p1">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c4">S2</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:152_3_1"><span class="lab"><font color="Green" title="E89">DD6</font></span></a>, <a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/><a NAME="E9:152_3_1_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">S2</font> <a href="fomodel1.html#K21" title="FOMODEL1:func.21">-multiCat</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c64">st11</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="#E7:152_3_1_2_2"><span class="lab"><font color="Green" title="E91">E1</font></span></a>, <a class="ref" href="fomodel0.html#T52" title="FOMODEL0:th.52">FOMODEL0:52</a></span>;<br/><a NAME="E10:152_3_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">E3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">S1</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="p1">(<span class="default"><a href="fomodel1.html#K33" title="FOMODEL1:func.33">SubTerms</a> <font color="Maroon" title="c47">t1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S2</font> <a href="fomodel1.html#K21" title="FOMODEL1:func.21">-multiCat</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K33" title="FOMODEL1:func.33">SubTerms</a> <font color="Maroon" title="c47">t1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T52" title="FOMODEL0:th.52">FOMODEL0:52</a></span>;<br/><a NAME="E11:152_3_1_2_2"/><span class="lab"><font color="Green" title="E93">E4</font></span>: 
<font color="Maroon" title="c47">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c61">t2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:152_3_1_2_2"><span class="lab"><font color="Green" title="E92">E3</font></span></a>, <a class="ref" href="fomodel1.html#D37" title="FOMODEL1:def.37">FOMODEL1:def 37</a></span>;<br/><a NAME="E12:152_3_1_2_2"/><span class="kw">then </span>
( <font color="Maroon" title="c47">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp; <font color="Maroon" title="c47">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S2</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="c35">NN</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">hence </span><a NAME="E13:152_3_1_2_2"/>
<font color="Maroon" title="c44">y</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"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c38">D2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:152_3"><span class="lab"><font color="Green" title="E77">C1</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <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="c27">I1</font> <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="c36">D1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">y</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="c28">I2</font> <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="c38">D2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">y</font></span><br/><a NAME="E14:152_3_1_2_2"/><span class="lab"><font color="Green" title="E94">DD15</font></span>: 
( <font color="Maroon" title="c61">t2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S2</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="c35">NN</font> &amp; <font color="Maroon" title="c61">t2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:152_3_1_2_2"><span class="lab"><font color="Green" title="E92">E3</font></span></a>, <a class="txt" href="#E7:152_3_1"><span class="lab"><font color="Green" title="E80">D9</font></span></a>, <a class="ref" href="fomodel1.html#D33" title="FOMODEL1:def.33">FOMODEL1:def 33</a>, <a class="ref" href="fomodel1.html#D37" title="FOMODEL1:def.37">FOMODEL1:def 37</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c65">D22</font> = <font color="Maroon" title="c38">D2</font> as   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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:152_3_1_2_2"><span class="lab"><font color="Green" title="E94">DD15</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c66">tt2</font> = <font color="Maroon" title="c61">t2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c65">D22</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:152_3_1_2_2"><span class="lab"><font color="Green" title="E94">DD15</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="lab"><font color="Green" title="E95">D12</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">S2</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="c61">t2</font> = 
<font color="Maroon" title="c61">t2</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/>.= 
<font color="Maroon" title="c54">o2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T41" title="FINSEQ_1:th.41">FINSEQ_1:41</a></span>
;<br/><a NAME="E18:152_3_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">D13</font></span>: 
<font color="Maroon" title="c60">st22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel1.html#K33" title="FOMODEL1:func.33">SubTerms</a> <font color="Maroon" title="c61">t2</font>
 <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="lab"><font color="Green" title="E97">D25</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c37">d1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c39">d2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c37">d1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:152_3"><span class="lab"><font color="Green" title="E75">C0</font></span></a>, <a class="txt" href="#E6:152_3"><span class="lab"><font color="Green" title="E77">C1</font></span></a>, <a class="ref" href="grfunc_1.html#T34" title="GRFUNC_1:th.34">GRFUNC_1:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <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"><font color="Maroon" title="c42">d12</font> <a href="fomodel0.html#K13" title="FOMODEL0:func.13">null</a> <font color="Maroon" title="c39">d2</font></span>)</span>
<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="E20:152_3_1_2_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c45">D11</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">y1</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="c27">I1</font> <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="c46">y1</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="c28">I2</font> <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="c65">D22</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c66">tt2</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="c28">I2</font> <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="c66">tt2</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="E21:152_3_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">D16</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c36">D1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c46">y1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c38">D2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c61">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c61">t2</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="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c36">D1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">I1</font> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c48">o1</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"><font color="Maroon" title="c27">I1</font> <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="c47">t1</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"><font color="Maroon" title="c27">I1</font> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c48">o1</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="c27">I1</font> <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="c37">d1</font></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="c47">t1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:152_3"><span class="lab"><font color="Green" title="E77">C1</font></span></a>, <a class="txt" href="#E27:152_3_1"><span class="lab"><font color="Green" title="E87">DD10</font></span></a>, <a class="txt" href="#E26:152_3_1"><span class="lab"><font color="Green" title="E86">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="c27">I1</font> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c48">o1</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"><font color="Maroon" title="c28">I2</font> <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="c47">t1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:152_3"><span class="lab"><font color="Green" title="E77">C1</font></span></a>, <a class="txt" href="#E27:152_3_1"><span class="lab"><font color="Green" title="E87">DD10</font></span></a>, <a class="txt" href="#E26:152_3_1"><span class="lab"><font color="Green" title="E86">D2</font></span></a>, <a class="txt" href="#E19:152_3_1_2_2"><span class="lab"><font color="Green" title="E97">D25</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"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c38">D2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:152_3_1_2_2"><span class="lab"><font color="Green" title="E93">E4</font></span></a>, <a class="txt" href="#E21:152_3_1_2_2"><span class="lab"><font color="Green" title="E98">D16</font></span></a>, <a class="txt" href="#E17:152_3_1_2_2"><span class="lab"><font color="Green" title="E95">D12</font></span></a>, <a class="txt" href="#E18:152_3_1_2_2"><span class="lab"><font color="Green" title="E96">D13</font></span></a>, <a class="txt" href="#E23:152_3_1"><span class="lab"><font color="Green" title="E85">D17</font></span></a>, <a class="ref" href="fomodel2.html#T21" title="FOMODEL2:th.21">FOMODEL2:21</a></span>
;<br/> <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>
<span class="kw">hence </span><a NAME="E11:152_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c33">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="fomodel0.html#T51" title="FOMODEL0:th.51">FOMODEL0:51</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="E8:152"/><span class="lab"><font color="Green" title="E75">B6</font></span>: 
 for <font color="Olive" title="b1">n</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">n</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="#E5:152"><span class="lab"><font color="Green" title="E72">B0</font></span></a>, <a class="txt" href="#E7:152"><span class="lab"><font color="Green" title="E74">B1</font></span></a>);<br/></span>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="152_4"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c33">g1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c27">I1</font> <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"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c34">g2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <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"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span>;<br/><a NAME="E1:152_4"/><span class="lab"><font color="Green" title="E76">C4</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"><font color="Maroon" title="c27">I1</font> <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="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c3">S1</font></span>)</span> &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"><font color="Maroon" title="c28">I2</font> <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="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c4">S2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:152"><span class="lab"><font color="Green" title="E70">B2</font></span></a>, <a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/><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>  <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="c27">I1</font> <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="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span></span>)</span> implies ( <font color="Maroon" title="c35">x</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"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span></span>)</span> <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="c28">I2</font> <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="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">x</font> ) )</span><br/><span class="kw">assume </span><a NAME="E2:152_4"/><span class="lab"><font color="Green" title="E77">C2</font></span>: 
<font color="Maroon" title="c35">x</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"><span class="p2">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></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="c35">x</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"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span></span>)</span> <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="c28">I2</font> <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="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">x</font> )</span><br/><a NAME="E3:152_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">C0</font></span>: 
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c3">S1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:152"><span class="lab"><font color="Green" title="E70">B2</font></span></a>, <a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c36">t1</font> = <font color="Maroon" title="c35">x</font> as   <a href="fomodel1.html#V13" title="FOMODEL1:attr.13">termal</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c3">S1</font> ;<br/><span class="kw">set </span><font color="Maroon" title="c37">m1</font> =  <a href="fomodel1.html#K38" title="FOMODEL1:func.38">Depth</a> <font color="Maroon" title="c36">t1</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c38">mm1</font> =  <a href="fomodel1.html#K38" title="FOMODEL1:func.38">Depth</a> <font color="Maroon" title="c36">t1</font> 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">reconsider </span><font color="Maroon" title="c39">t11</font> = <font color="Maroon" title="c36">t1</font> as   <a href="fomodel1.html#K38" title="FOMODEL1:func.38">Depth</a> <font color="Maroon" title="c36">t1</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="c3">S1</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/><a NAME="E7:152_4"/><span class="lab"><font color="Green" title="E79">C6</font></span>: 
( <font color="Maroon" title="c39">t11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">S1</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> <span class="p1">(<span class="default"><a href="fomodel1.html#K38" title="FOMODEL1:func.38">Depth</a> <font color="Maroon" title="c36">t1</font></span>)</span> &amp; <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:152_4"><span class="lab"><font color="Green" title="E78">C0</font></span></a>, <a class="ref" href="fomodel1.html#D33" title="FOMODEL1:def.33">FOMODEL1:def 33</a></span>;<br/><a NAME="E8:152_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">C1</font></span>: 
( <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></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"><font color="Maroon" title="c3">S1</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> <span class="p2">(<span class="default"><a href="fomodel1.html#K38" title="FOMODEL1:func.38">Depth</a> <font color="Maroon" title="c36">t1</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">S1</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="c38">mm1</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="c3">S1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">S2</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="c38">mm1</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">S2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel1.html#T2" title="FOMODEL1:th.2">FOMODEL1:2</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c40">D1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></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"><font color="Maroon" title="c3">S1</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> <span class="p2">(<span class="default"><a href="fomodel1.html#K38" title="FOMODEL1:func.38">Depth</a> <font color="Maroon" title="c36">t1</font></span>)</span></span>)</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#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c3">S1</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/><span class="kw">reconsider </span><font color="Maroon" title="c41">D2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></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"><font color="Maroon" title="c4">S2</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> <span class="p2">(<span class="default"><a href="fomodel1.html#K38" title="FOMODEL1:func.38">Depth</a> <font color="Maroon" title="c36">t1</font></span>)</span></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">S2</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:152_4"><span class="lab"><font color="Green" title="E80">C1</font></span></a>, <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="c42">t111</font> = <font color="Maroon" title="c36">t1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c40">D1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:152_4"><span class="lab"><font color="Green" title="E79">C6</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c43">f1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c40">D1</font>;<br/><span class="kw">set </span><font color="Maroon" title="c44">f2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c41">D2</font>;<br/><a NAME="E12:152_4"/><span class="lab"><font color="Green" title="E81">C5</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"><font color="Maroon" title="c27">I1</font> <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="c40">D1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">D1</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"><font color="Maroon" title="c28">I2</font> <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="c41">D2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">D2</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="E13:152_4"/>
( <font color="Maroon" title="c35">x</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"><span class="p2">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c40">D1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c40">D1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c41">D2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:152"><span class="lab"><font color="Green" title="E75">B6</font></span></a>, <a class="txt" href="#E8:152_4"><span class="lab"><font color="Green" title="E80">C1</font></span></a>, <a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/><a NAME="E14:152_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">C3</font></span>: 
( <font color="Maroon" title="c35">x</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"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="c41">D2</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c40">D1</font></span>)</span> <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="c28">I2</font> <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="c41">D2</font></span>)</span> <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="ref" href="fomodel0.html#T51" title="FOMODEL0:th.51">FOMODEL0:51</a></span>;<br/><a NAME="E15:152_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">C9</font></span>: 
( <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c4">S2</font></span>)</span> &amp; <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">D2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:152_4"><span class="lab"><font color="Green" title="E81">C5</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E16:152_4"/>
<font color="Maroon" title="c35">x</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"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:152"><span class="lab"><font color="Green" title="E70">B2</font></span></a>, <a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>; <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="c27">I1</font> <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="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span></span>)</span> <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="c28">I2</font> <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="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">x</font></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span></span>)</span> <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"><font color="Maroon" title="c27">I1</font> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c42">t111</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:152_4"><span class="lab"><font color="Green" title="E77">C2</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">I1</font> <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="c40">D1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">t111</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:152_4"><span class="lab"><font color="Green" title="E81">C5</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <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">t111</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:152_4"><span class="lab"><font color="Green" title="E82">C3</font></span></a>, <a class="txt" href="#E12:152_4"><span class="lab"><font color="Green" title="E81">C5</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">I2</font> <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="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span></span>)</span> <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="#E15:152_4"><span class="lab"><font color="Green" title="E83">C9</font></span></a>, <a class="txt" href="#E1:152_4"><span class="lab"><font color="Green" title="E76">C4</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</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>
<span class="kw">hence </span><a NAME="E10:152"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c27">I1</font> <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"><font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">I2</font> <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"><font color="Maroon" title="c1">X</font> <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="fomodel0.html#T51" title="FOMODEL0:th.51">FOMODEL0:51</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/>


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