<?xml version="1.0"?>
<div class="add">

<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">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">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#K33" title="FOMODEL2:func.33">AllFormulasOf</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;  not <font color="Maroon" title="c1">X</font> is <font color="Olive" title="b2">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  holds <br/> not (<font color="Olive" title="b2">D</font>,<font color="Olive" title="b3">num</font>) <a href="fomodel4.html#K49" title="FOMODEL4:func.49">CompletionOf</a> <font color="Maroon" title="c1">X</font> is <font color="Olive" title="b2">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</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">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#K33" title="FOMODEL2:func.33">AllFormulasOf</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;  not <font color="Maroon" title="c1">X</font> is <font color="Olive" title="b1">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  holds <br/> not (<font color="Olive" title="b1">D</font>,<font color="Olive" title="b2">num</font>) <a href="fomodel4.html#K49" title="FOMODEL4:func.49">CompletionOf</a> <font color="Maroon" title="c1">X</font> is <font color="Olive" title="b1">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</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">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#K33" title="FOMODEL2:func.33">AllFormulasOf</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;  not <font color="Maroon" title="c1">X</font> is <font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  holds <br/> not (<font color="Maroon" title="c3">D</font>,<font color="Olive" title="b1">num</font>) <a href="fomodel4.html#K49" title="FOMODEL4:func.49">CompletionOf</a> <font color="Maroon" title="c1">X</font> is <font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a> </span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">EF</font> =  <a href="fomodel2.html#K43" title="FOMODEL2:func.43">ExFormulasOf</a> <font color="Maroon" title="c2">S</font>;<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">FF</font> =  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">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#K33" title="FOMODEL2:func.33">AllFormulasOf</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;  not <font color="Maroon" title="c1">X</font> is <font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  implies  not (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K49" title="FOMODEL4:func.49">CompletionOf</a> <font color="Maroon" title="c1">X</font> is <font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">XX</font> = (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K49" title="FOMODEL4:func.49">CompletionOf</a> <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">G1</font> =  <a href="fomodel4.html#K23" title="FOMODEL4:func.23">R1</a> <font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">G8</font> =  <a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c2">S</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/>
<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/>
<span class="kw">set </span><font color="Maroon" title="c13">f</font> = (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">assume </span><a NAME="E1:251"/><span class="lab"><font color="Green" title="E136">B0</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;  not <font color="Maroon" title="c1">X</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 (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K49" title="FOMODEL4:func.49">CompletionOf</a> <font color="Maroon" title="c1">X</font> is <font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a> </span><br/>

<a NAME="E2:251"/><span class="lab"><font color="Green" title="E137">B1</font></span>: 
 for <font color="Olive" title="b1">nn</font>, <font color="Olive" title="b2">mm</font> being    <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>   st <font color="Olive" title="b2">mm</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Olive" title="b1">nn</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Olive" title="b1">nn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">mm</font> holds <br/><span class="p1">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">nn</font> <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="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">mm</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="251_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">nn</font>, <font color="Maroon" title="c15">mm</font> be    <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> ; <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="c15">mm</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Maroon" title="c14">nn</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Maroon" title="c14">nn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">mm</font> implies <span class="p1">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">nn</font> <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="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">mm</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c16">m</font> = <font color="Maroon" title="c15">mm</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">n</font> = <font color="Maroon" title="c14">nn</font>;<br/>
<span class="kw">assume </span><a NAME="E1:251_1"/>
( <font color="Maroon" title="c15">mm</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Maroon" title="c14">nn</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Maroon" title="c14">nn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">mm</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="c3">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">nn</font> <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="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">mm</font></span><br/>

<a NAME="E2:251_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">nn</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">nn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">mm</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">nn</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E3:251_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">mm</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">nn</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">k</font> = <font color="Maroon" title="c15">mm</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">nn</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<a NAME="E5:251_1"/>
<span class="p1">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">nn</font> <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="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c1">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="c14">nn</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:251"><span class="lab"><font color="Green" title="E136">B0</font></span></a>, <a class="txt" href="fomodel4.html#E164"><span class="lab"><font color="Green" title="E133">Lm4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:251_1"/>
<span class="p1">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">nn</font> <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="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">mm</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="251_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">Y</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K49" title="FOMODEL4:func.49">CompletionOf</a> <font color="Maroon" title="c1">X</font></span>)</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="c14">Y</font> is <font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a> </span><br/><span class="kw">consider </span><font color="Maroon" title="c15">kk</font> being    <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"> such that </span><br/><a NAME="E2:251_2"/><span class="lab"><font color="Green" title="E138">C0</font></span>: 
<font color="Maroon" title="c14">Y</font> <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="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">kk</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:251"><span class="lab"><font color="Green" title="E137">B1</font></span></a>, <a class="ref" href="henmodel.html#T3" title="HENMODEL:th.3">HENMODEL:3</a></span>;<br/><a NAME="E3:251_2"/>
 not <span class="p1">(<span class="default">(<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">kk</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="#E1:251"><span class="lab"><font color="Green" title="E136">B0</font></span></a>, <a class="txt" href="fomodel4.html#E164"><span class="lab"><font color="Green" title="E133">Lm4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:251_2"/>
 not <font color="Maroon" title="c14">Y</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="#E2:251_2"><span class="lab"><font color="Green" title="E138">C0</font></span></a>, <a class="ref" href="fomodel4.html#T15" target="_self" title="FOMODEL4:th.15">Th16</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="E4:251"/>
 not (<font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K49" title="FOMODEL4:func.49">CompletionOf</a> <font color="Maroon" title="c1">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="txt" href="fomodel4.html#E136"><span class="lab"><font color="Green" title="E106">Lm1</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"> verum</span><br/>


</div>
