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<span class="kw">let </span><font color="Maroon" title="c1">Z</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">S</font> being   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a><br/>  for <font color="Olive" title="b2">D</font> being   <a href="fomodel4.html#NM2" title="FOMODEL4:NM.2">RuleSet</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b3">X</font> being   <a href="funct_1.html#V4" title="FUNCT_1:attr.4">functional</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b4">num</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p1">(<span class="default"><a href="fomodel2.html#K43" title="FOMODEL2:func.43">ExFormulasOf</a> <font color="Olive" title="b1">S</font></span>)</span>  st <font color="Olive" title="b2">D</font> is  <a href="fomodel4.html#V6" title="FOMODEL4:attr.6">isotone</a>  &amp;  <a href="fomodel4.html#K23" title="FOMODEL4:func.23">R1</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">D</font> &amp;  <a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">D</font> &amp;  <a href="fomodel4.html#K24" title="FOMODEL4:func.24">R2</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">D</font> &amp;  <a href="fomodel4.html#K30" title="FOMODEL4:func.30">R5</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">D</font> &amp;  not <span class="p1">(<span class="default"><a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="fomodel0.html#K21" title="FOMODEL0:func.21">SymbolsOf</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="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="p4"><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>)</span></span>)</span></span>)</span> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  &amp; <font color="Olive" title="b3">X</font> <a href="fomodel4.html#NK47" title="FOMODEL4:NK.47">addw</a> (<font color="Olive" title="b2">D</font>,<font color="Olive" title="b4">num</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Z</font> &amp;  not <font color="Maroon" title="c1">Z</font> is <font color="Olive" title="b2">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b4">num</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel2.html#K43" title="FOMODEL2:func.43">ExFormulasOf</a> <font color="Olive" title="b1">S</font> holds <br/><font color="Maroon" title="c1">Z</font> is <font color="Olive" title="b1">S</font> <a href="fomodel4.html#V15" title="FOMODEL4:attr.15">-witnessed</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">D</font> being   <a href="fomodel4.html#NM2" title="FOMODEL4:NM.2">RuleSet</a> of <font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b2">X</font> being   <a href="funct_1.html#V4" title="FUNCT_1:attr.4">functional</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b3">num</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p1">(<span class="default"><a href="fomodel2.html#K43" title="FOMODEL2:func.43">ExFormulasOf</a> <font color="Maroon" title="c2">S</font></span>)</span>  st <font color="Olive" title="b1">D</font> is  <a href="fomodel4.html#V6" title="FOMODEL4:attr.6">isotone</a>  &amp;  <a href="fomodel4.html#K23" title="FOMODEL4:func.23">R1</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font> &amp;  <a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font> &amp;  <a href="fomodel4.html#K24" title="FOMODEL4:func.24">R2</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font> &amp;  <a href="fomodel4.html#K30" title="FOMODEL4:func.30">R5</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font> &amp;  not <span class="p1">(<span class="default"><a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="fomodel0.html#K21" title="FOMODEL0:func.21">SymbolsOf</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c2">S</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="p4"><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>)</span></span>)</span></span>)</span> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  &amp; <font color="Olive" title="b2">X</font> <a href="fomodel4.html#NK47" title="FOMODEL4:NK.47">addw</a> (<font color="Olive" title="b1">D</font>,<font color="Olive" title="b3">num</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Z</font> &amp;  not <font color="Maroon" title="c1">Z</font> is <font color="Olive" title="b1">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b3">num</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel2.html#K43" title="FOMODEL2:func.43">ExFormulasOf</a> <font color="Maroon" title="c2">S</font> holds <br/><font color="Maroon" title="c1">Z</font> is <font color="Maroon" title="c2">S</font> <a href="fomodel4.html#V15" title="FOMODEL4:attr.15">-witnessed</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">D</font> be   <a href="fomodel4.html#NM2" title="FOMODEL4:NM.2">RuleSet</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being   <a href="funct_1.html#V4" title="FUNCT_1:attr.4">functional</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b2">num</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p1">(<span class="default"><a href="fomodel2.html#K43" title="FOMODEL2:func.43">ExFormulasOf</a> <font color="Maroon" title="c2">S</font></span>)</span>  st <font color="Maroon" title="c3">D</font> is  <a href="fomodel4.html#V6" title="FOMODEL4:attr.6">isotone</a>  &amp;  <a href="fomodel4.html#K23" title="FOMODEL4:func.23">R1</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp;  <a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp;  <a href="fomodel4.html#K24" title="FOMODEL4:func.24">R2</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp;  <a href="fomodel4.html#K30" title="FOMODEL4:func.30">R5</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp;  not <span class="p1">(<span class="default"><a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="fomodel0.html#K21" title="FOMODEL0:func.21">SymbolsOf</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c2">S</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="p4"><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>)</span></span>)</span></span>)</span> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  &amp; <font color="Olive" title="b1">X</font> <a href="fomodel4.html#NK47" title="FOMODEL4:NK.47">addw</a> (<font color="Maroon" title="c3">D</font>,<font color="Olive" title="b2">num</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Z</font> &amp;  not <font color="Maroon" title="c1">Z</font> is <font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">num</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel2.html#K43" title="FOMODEL2:func.43">ExFormulasOf</a> <font color="Maroon" title="c2">S</font> holds <br/><font color="Maroon" title="c1">Z</font> is <font color="Maroon" title="c2">S</font> <a href="fomodel4.html#V15" title="FOMODEL4:attr.15">-witnessed</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">X</font> be   <a href="funct_1.html#V4" title="FUNCT_1:attr.4">functional</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">num</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p1">(<span class="default"><a href="fomodel2.html#K43" title="FOMODEL2:func.43">ExFormulasOf</a> <font color="Maroon" title="c2">S</font></span>)</span>  st <font color="Maroon" title="c3">D</font> is  <a href="fomodel4.html#V6" title="FOMODEL4:attr.6">isotone</a>  &amp;  <a href="fomodel4.html#K23" title="FOMODEL4:func.23">R1</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp;  <a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp;  <a href="fomodel4.html#K24" title="FOMODEL4:func.24">R2</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp;  <a href="fomodel4.html#K30" title="FOMODEL4:func.30">R5</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp;  not <span class="p1">(<span class="default"><a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="fomodel0.html#K21" title="FOMODEL0:func.21">SymbolsOf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c2">S</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="p4"><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>)</span></span>)</span></span>)</span> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  &amp; <font color="Maroon" title="c4">X</font> <a href="fomodel4.html#NK47" title="FOMODEL4:NK.47">addw</a> (<font color="Maroon" title="c3">D</font>,<font color="Olive" title="b1">num</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Z</font> &amp;  not <font color="Maroon" title="c1">Z</font> is <font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">num</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel2.html#K43" title="FOMODEL2:func.43">ExFormulasOf</a> <font color="Maroon" title="c2">S</font> holds <br/><font color="Maroon" title="c1">Z</font> is <font color="Maroon" title="c2">S</font> <a href="fomodel4.html#V15" title="FOMODEL4:attr.15">-witnessed</a> </span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">L</font> =  <a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">F</font> = <font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">EF</font> =  <a href="fomodel2.html#K43" title="FOMODEL2:func.43">ExFormulasOf</a> <font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">num</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p1">(<span class="default"><a href="fomodel2.html#K43" title="FOMODEL2:func.43">ExFormulasOf</a> <font color="Maroon" title="c2">S</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="c3">D</font> is  <a href="fomodel4.html#V6" title="FOMODEL4:attr.6">isotone</a>  &amp;  <a href="fomodel4.html#K23" title="FOMODEL4:func.23">R1</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp;  <a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp;  <a href="fomodel4.html#K24" title="FOMODEL4:func.24">R2</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp;  <a href="fomodel4.html#K30" title="FOMODEL4:func.30">R5</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp;  not <span class="p1">(<span class="default"><a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="fomodel0.html#K21" title="FOMODEL0:func.21">SymbolsOf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c2">S</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="p4"><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>)</span></span>)</span></span>)</span> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  &amp; <font color="Maroon" title="c4">X</font> <a href="fomodel4.html#NK47" title="FOMODEL4:NK.47">addw</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Z</font> &amp;  not <font color="Maroon" title="c1">Z</font> is <font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">num</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel2.html#K43" title="FOMODEL2:func.43">ExFormulasOf</a> <font color="Maroon" title="c2">S</font> implies <font color="Maroon" title="c1">Z</font> is <font color="Maroon" title="c2">S</font> <a href="fomodel4.html#V15" title="FOMODEL4:attr.15">-witnessed</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c9">f</font> = (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">Y</font> = <font color="Maroon" title="c4">X</font> <a href="fomodel4.html#NK47" title="FOMODEL4:NK.47">addw</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c11">SS</font> =  <a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font>;<br/>
<a NAME="E1:242"/>
<font color="Maroon" title="c4">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="fomodel4.html#NK47" title="FOMODEL4:NK.47">addw</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</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="E2:242"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">B9</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> <a href="fomodel4.html#NK47" title="FOMODEL4:NK.47">addw</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:242"/><span class="lab"><font color="Green" title="E132">B8</font></span>: 
( <font color="Maroon" title="c3">D</font> is  <a href="fomodel4.html#V6" title="FOMODEL4:attr.6">isotone</a>  &amp;  <a href="fomodel4.html#K23" title="FOMODEL4:func.23">R1</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp;  <a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp;  <a href="fomodel4.html#K24" title="FOMODEL4:func.24">R2</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp;  <a href="fomodel4.html#K30" title="FOMODEL4:func.30">R5</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp;  not <span class="p1">(<span class="default"><a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="fomodel0.html#K21" title="FOMODEL0:func.21">SymbolsOf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c2">S</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="p4"><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>)</span></span>)</span></span>)</span> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">X</font> <a href="fomodel4.html#NK47" title="FOMODEL4:NK.47">addw</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Z</font> or <font color="Maroon" title="c1">Z</font> is <font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  or  not  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">num</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel2.html#K43" title="FOMODEL2:func.43">ExFormulasOf</a> <font color="Maroon" title="c2">S</font> or <font color="Maroon" title="c1">Z</font> is <font color="Maroon" title="c2">S</font> <a href="fomodel4.html#V15" title="FOMODEL4:attr.15">-witnessed</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E4:242"/><span class="lab"><font color="Green" title="E133">B0</font></span>: 
( <font color="Maroon" title="c4">X</font> <a href="fomodel4.html#NK47" title="FOMODEL4:NK.47">addw</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Z</font> &amp;  not <font color="Maroon" title="c1">Z</font> is <font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">num</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel2.html#K43" title="FOMODEL2:func.43">ExFormulasOf</a> <font color="Maroon" title="c2">S</font> or <font color="Maroon" title="c1">Z</font> is <font color="Maroon" title="c2">S</font> <a href="fomodel4.html#V15" title="FOMODEL4:attr.15">-witnessed</a>  )</span><br/>

<a NAME="E5:242"/><span class="kw">then </span>
( <font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Z</font> &amp;  not <font color="Maroon" title="c1">Z</font> is <font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:242"><span class="lab"><font color="Green" title="E131">B9</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E6:242"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">B20</font></span>: 
 not <font color="Maroon" title="c4">X</font> is <font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#T15" target="_self" title="FOMODEL4:th.15">Th16</a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:242"/><span class="lab"><font color="Green" title="E135">B1</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">num</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel2.html#K43" title="FOMODEL2:func.43">ExFormulasOf</a> <font color="Maroon" title="c2">S</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="c1">Z</font> is <font color="Maroon" title="c2">S</font> <a href="fomodel4.html#V15" title="FOMODEL4:attr.15">-witnessed</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c12">strings</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="c2">S</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/>
<a NAME="E8:242"/>
 for <font color="Olive" title="b1">l1</font> being   <a href="fomodel1.html#V4" title="FOMODEL1:attr.4">literal</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b2">phi1</font> being   <a href="fomodel2.html#V4" title="FOMODEL2:attr.4">wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</font>  st <span class="p1"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><font color="Olive" title="b1">l1</font></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Olive" title="b2">phi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font> holds <br/> ex <font color="Olive" title="b3">l2</font> being   <a href="fomodel1.html#V4" title="FOMODEL1:attr.4">literal</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( (<font color="Olive" title="b1">l1</font>,<font color="Olive" title="b3">l2</font>) <a href="fomodel3.html#K24" title="FOMODEL3:func.24">-SymbolSubstIn</a> <font color="Olive" title="b2">phi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font> &amp;  not <font color="Olive" title="b3">l2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">phi1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="242_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">l1</font> be   <a href="fomodel1.html#V4" title="FOMODEL1:attr.4">literal</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">phi1</font> being   <a href="fomodel2.html#V4" title="FOMODEL2:attr.4">wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</font>  st <span class="p1"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><font color="Maroon" title="c13">l1</font></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Olive" title="b1">phi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font> holds <br/> ex <font color="Olive" title="b2">l2</font> being   <a href="fomodel1.html#V4" title="FOMODEL1:attr.4">literal</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( (<font color="Maroon" title="c13">l1</font>,<font color="Olive" title="b2">l2</font>) <a href="fomodel3.html#K24" title="FOMODEL3:func.24">-SymbolSubstIn</a> <font color="Olive" title="b1">phi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font> &amp;  not <font color="Olive" title="b2">l2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">phi1</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c14">phi1</font> be   <a href="fomodel2.html#V4" title="FOMODEL2:attr.4">wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</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"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><font color="Maroon" title="c13">l1</font></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c14">phi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font> implies  ex <font color="Olive" title="b1">l2</font> being   <a href="fomodel1.html#V4" title="FOMODEL1:attr.4">literal</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( (<font color="Maroon" title="c13">l1</font>,<font color="Olive" title="b1">l2</font>) <a href="fomodel3.html#K24" title="FOMODEL3:func.24">-SymbolSubstIn</a> <font color="Maroon" title="c14">phi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font> &amp;  not <font color="Olive" title="b1">l2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phi1</font> ) )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c15">phi</font> = <span class="p1"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><font color="Maroon" title="c13">l1</font></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c14">phi1</font>;<br/>
<a NAME="E1:242_1"/>
( <span class="p1"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><font color="Maroon" title="c13">l1</font></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c14">phi1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><font color="Maroon" title="c13">l1</font></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c14">phi1</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> &amp;  not <span class="p1"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><font color="Maroon" title="c13">l1</font></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c14">phi1</font> is  <a href="fomodel1.html#V15" title="FOMODEL1:attr.15">0wff</a>  )
 
;<br/>
<a NAME="E2:242_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E136">B3</font></span>: 
( <font color="Maroon" title="c13">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><font color="Maroon" title="c13">l1</font></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c14">phi1</font></span>)</span> &amp; <font color="Maroon" title="c14">phi1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel2.html#K40" title="FOMODEL2:func.40">head</a> <span class="p1">(<span class="default"><span class="p2"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><font color="Maroon" title="c13">l1</font></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c14">phi1</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel2.html#T23" title="FOMODEL2:th.23">FOMODEL2:23</a></span>;<br/>
<a NAME="E3:242_1"/>
<span class="p1"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><font color="Maroon" title="c13">l1</font></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c14">phi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel2.html#K43" title="FOMODEL2:func.43">ExFormulasOf</a> <font color="Maroon" title="c2">S</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">phii</font> = <span class="p1"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><font color="Maroon" title="c13">l1</font></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c14">phi1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="fomodel2.html#K43" title="FOMODEL2:func.43">ExFormulasOf</a> <font color="Maroon" title="c2">S</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:242_1"/><span class="lab"><font color="Green" title="E137">C0</font></span>: 
( <font color="Maroon" title="c17">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> <font color="Maroon" title="c8">num</font> &amp; <font color="Maroon" title="c8">num</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">phii</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:242"><span class="lab"><font color="Green" title="E135">B1</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">mm</font> = <font color="Maroon" title="c17">x</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="txt" href="#E6:242_1"><span class="lab"><font color="Green" title="E137">C0</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">MM</font> = <font color="Maroon" title="c18">mm</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">Xm</font> = <span class="p1">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">mm</font> as   <a href="funct_1.html#V4" title="FUNCT_1:attr.4">functional</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">no</font> =  <a href="fomodel0.html#K21" title="FOMODEL0:func.21">SymbolsOf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c2">S</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="p3"><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>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">mm</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">phi1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">T</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c2">S</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="p2"><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>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">phi1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="finseq_1.html#V3" title="FINSEQ_1:attr.3">FinSequence-membered</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">phi1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">t</font> =  <a href="fomodel0.html#K21" title="FOMODEL0:func.21">SymbolsOf</a> <font color="Maroon" title="c22">T</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">i</font> = <span class="p1">(<span class="default"><a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="fomodel0.html#K21" title="FOMODEL0:func.21">SymbolsOf</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">mm</font></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"><span class="p5">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c2">S</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="p4"><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>)</span></span>)</span></span>)</span> as   <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#T17" target="_self" title="FOMODEL4:th.17">Th18</a>, <a class="txt" href="#E3:242"><span class="lab"><font color="Green" title="E132">B8</font></span></a>, <a class="txt" href="#E6:242"><span class="lab"><font color="Green" title="E134">B20</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E138">B6</font></span>:  <a href="fomodel0.html#K21" title="FOMODEL0:func.21">SymbolsOf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c2">S</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="p3"><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>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">mm</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">phi1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> = 
 <a href="fomodel0.html#K21" title="FOMODEL0:func.21">SymbolsOf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c2">S</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="p4"><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>)</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">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">mm</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c2">S</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="p4"><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>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">phi1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fomodel0.html#K21" title="FOMODEL0:func.21">SymbolsOf</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c2">S</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="p4"><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>)</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">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">mm</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="fomodel0.html#K21" title="FOMODEL0:func.21">SymbolsOf</a> <font color="Maroon" title="c22">T</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T47" title="FOMODEL0:th.47">FOMODEL0:47</a></span>
;<br/>
<a NAME="E14:242_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="fomodel0.html#K21" title="FOMODEL0:func.21">SymbolsOf</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c2">S</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="p4"><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>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">mm</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">phi1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">i</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c23">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T41" title="XBOOLE_1:th.41">XBOOLE_1:41</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c25">yes</font> = <span class="p1">(<span class="default"><a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="fomodel0.html#K21" title="FOMODEL0:func.21">SymbolsOf</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c2">S</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="p4"><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>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">mm</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">phi1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></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#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c2">S</font></span>)</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c26">ll2</font> =  the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c25">yes</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c27">l2</font> =  the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c25">yes</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="c2">S</font> <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">set </span><font color="Maroon" title="c28">psi1</font> = (<font color="Maroon" title="c13">l1</font>,<font color="Maroon" title="c27">l2</font>) <a href="fomodel3.html#K24" title="FOMODEL3:func.24">-SymbolSubstIn</a> <font color="Maroon" title="c14">phi1</font>;<br/>
<a NAME="E17:242_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<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="E18:242_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E139">B4</font></span>: 
( <span class="p1">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">mm</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font></span>)</span> &amp; <span class="p1">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c19">MM</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E19:242_1"/><span class="kw">then </span>
<span class="p1">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">mm</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> <a href="fomodel4.html#NK47" title="FOMODEL4:NK.47">addw</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/>
<a NAME="E20:242_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">B5</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">mm</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:242"><span class="lab"><font color="Green" title="E133">B0</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">assume </span><a NAME="E21:242_1"/>
<span class="p1"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><font color="Maroon" title="c13">l1</font></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c14">phi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">l2</font> being   <a href="fomodel1.html#V4" title="FOMODEL1:attr.4">literal</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( (<font color="Maroon" title="c13">l1</font>,<font color="Olive" title="b1">l2</font>) <a href="fomodel3.html#K24" title="FOMODEL3:func.24">-SymbolSubstIn</a> <font color="Maroon" title="c14">phi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font> &amp;  not <font color="Olive" title="b1">l2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phi1</font> )</span><br/>

<a NAME="E22:242_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><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="c13">l1</font></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c14">phi1</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<a NAME="E23:242_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">mm</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><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="c13">l1</font></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c14">phi1</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:242_1"><span class="lab"><font color="Green" title="E140">B5</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E24:242_1"/><span class="kw">then </span>
 not <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">mm</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><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="c13">l1</font></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c14">phi1</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:242"><span class="lab"><font color="Green" title="E133">B0</font></span></a>, <a class="ref" href="fomodel4.html#T15" target="_self" title="FOMODEL4:th.15">Th16</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">mm</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c13">l1</font>,<font color="Maroon" title="c27">l2</font>) <a href="fomodel3.html#K24" title="FOMODEL3:func.24">-SymbolSubstIn</a> <font color="Maroon" title="c14">phi1</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> = 
(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c16">phii</font>) <a href="fomodel4.html#K43" title="FOMODEL4:func.43">AddAsWitnessTo</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">mm</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D67" target="_self" title="FOMODEL4:def.67">DefWit2</a>, <a class="txt" href="#E2:242_1"><span class="lab"><font color="Green" title="E136">B3</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">mm</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D71" target="_self" title="FOMODEL4:def.71">DefWit3</a>, <a class="txt" href="#E6:242_1"><span class="lab"><font color="Green" title="E137">C0</font></span></a></span>
;<br/>
<a NAME="E26:242_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c13">l1</font>,<font color="Maroon" title="c27">l2</font>) <a href="fomodel3.html#K24" title="FOMODEL3:func.24">-SymbolSubstIn</a> <font color="Maroon" title="c14">phi1</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="fomodel0.html#K13" title="FOMODEL0:func.13">null</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">mm</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">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c19">MM</font>
 
;<br/>
<a NAME="E27:242_1"/><span class="kw">then </span>
(<font color="Maroon" title="c13">l1</font>,<font color="Maroon" title="c27">l2</font>) <a href="fomodel3.html#K24" title="FOMODEL3:func.24">-SymbolSubstIn</a> <font color="Maroon" title="c14">phi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c19">MM</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<a NAME="E28:242_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E141">B7</font></span>: 
(<font color="Maroon" title="c13">l1</font>,<font color="Maroon" title="c27">l2</font>) <a href="fomodel3.html#K24" title="FOMODEL3:func.24">-SymbolSubstIn</a> <font color="Maroon" title="c14">phi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="fomodel4.html#NK47" title="FOMODEL4:NK.47">addw</a> (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:242_1"><span class="lab"><font color="Green" title="E139">B4</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c27">l2</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="c13">l1</font>,<font color="Maroon" title="c27">l2</font>) <a href="fomodel3.html#K24" title="FOMODEL3:func.24">-SymbolSubstIn</a> <font color="Maroon" title="c14">phi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font> &amp;  not <font color="Maroon" title="c27">l2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phi1</font> )</span><br/>

<span class="kw">thus </span><a NAME="E29:242_1"/>
(<font color="Maroon" title="c13">l1</font>,<font color="Maroon" title="c27">l2</font>) <a href="fomodel3.html#K24" title="FOMODEL3:func.24">-SymbolSubstIn</a> <font color="Maroon" title="c14">phi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:242"><span class="lab"><font color="Green" title="E133">B0</font></span></a>, <a class="txt" href="#E28:242_1"><span class="lab"><font color="Green" title="E141">B7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c27">l2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phi1</font></span><br/>

<a NAME="E30:242_1"/>
 not <font color="Maroon" title="c27">l2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel0.html#K21" title="FOMODEL0:func.21">SymbolsOf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c2">S</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="p3"><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>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c8">num</font>) <a href="fomodel4.html#NK45" title="FOMODEL4:NK.45">addw</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">mm</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">phi1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E31:242_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c27">l2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel0.html#K21" title="FOMODEL0:func.21">SymbolsOf</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">phi1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:242_1"><span class="lab"><font color="Green" title="E138">B6</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E32:242_1"/>
 not <font color="Maroon" title="c27">l2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phi1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T45" title="FOMODEL0:th.45">FOMODEL0:45</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="E9:242"/>
<font color="Maroon" title="c1">Z</font> is <font color="Maroon" title="c2">S</font> <a href="fomodel4.html#V15" title="FOMODEL4:attr.15">-witnessed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D66" target="_self" title="FOMODEL4:def.66">DefWit1</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>
