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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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> holds <br/> (  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>  iff  for <font color="Olive" title="b3">Y</font> being   <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 <font color="Maroon" title="c1">X</font>  holds  not <font color="Olive" title="b3">Y</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> holds <br/> (  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>  iff  for <font color="Olive" title="b2">Y</font> being   <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 <font color="Maroon" title="c1">X</font>  holds  not <font color="Olive" title="b2">Y</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"> (  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>  iff  for <font color="Olive" title="b1">Y</font> being   <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 <font color="Maroon" title="c1">X</font>  holds  not <font color="Olive" title="b1">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">set </span><font color="Maroon" title="c4">N</font> =  <a href="fomodel2.html#K1" title="FOMODEL2:func.1">TheNorSymbOf</a> <font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">thus </span><a NAME="E1:192"/>
(  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  for <font color="Olive" title="b1">Y</font> being   <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 <font color="Maroon" title="c1">X</font>  holds  not <font color="Olive" title="b1">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="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"> ( (  for <font color="Olive" title="b1">Y</font> being   <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 <font color="Maroon" title="c1">X</font>  holds  not <font color="Olive" title="b1">Y</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="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">assume </span><a NAME="E2:192"/><span class="lab"><font color="Green" title="E107">C4</font></span>: 
 for <font color="Olive" title="b1">Y</font> being   <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 <font color="Maroon" title="c1">X</font>  holds  not <font color="Olive" title="b1">Y</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="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">assume </span><a NAME="E3:192"/>
<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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">phi1</font>, <font color="Maroon" title="c6">phi2</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><span class="kw"> such that </span><br/><a NAME="E5:192"/><span class="lab"><font color="Green" title="E108">C0</font></span>: 
( <font color="Maroon" title="c5">phi1</font> is <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a>  &amp; <span class="p1">(<span class="default"><span class="p2"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="fomodel2.html#K1" title="FOMODEL2:func.1">TheNorSymbOf</a> <font color="Maroon" title="c2">S</font></span>)</span></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="c5">phi1</font></span>)</span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c6">phi2</font> is <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D65" target="_self" title="FOMODEL4:def.65">DefInc</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">phi</font> = <span class="p1">(<span class="default"><span class="p2"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="fomodel2.html#K1" title="FOMODEL2:func.1">TheNorSymbOf</a> <font color="Maroon" title="c2">S</font></span>)</span></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="c5">phi1</font></span>)</span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c6">phi2</font> as   non  <a href="fomodel1.html#V15" title="FOMODEL1:attr.15">0wff</a>   <a href="fomodel2.html#V4" title="FOMODEL2:attr.4">wff</a>   non  <a href="fomodel2.html#V5" title="FOMODEL2:attr.5">exal</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">H1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c9">m1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:192"/><span class="lab"><font color="Green" title="E109">C1</font></span>: 
( <font color="Maroon" title="c8">H1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">H1</font>,<font color="Maroon" title="c5">phi1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is <font color="Maroon" title="c9">m1</font>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V4" title="FOMODEL4:attr.4">-derivable</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D62" target="_self" title="FOMODEL4:def.62">DefProvable3</a>, <a class="txt" href="#E5:192"><span class="lab"><font color="Green" title="E108">C0</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">H2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c11">m2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E10:192"/><span class="lab"><font color="Green" title="E110">C2</font></span>: 
( <font color="Maroon" title="c10">H2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">H2</font>,<font color="Maroon" title="c7">phi</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is <font color="Maroon" title="c11">m2</font>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V4" title="FOMODEL4:attr.4">-derivable</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D62" target="_self" title="FOMODEL4:def.62">DefProvable3</a>, <a class="txt" href="#E5:192"><span class="lab"><font color="Green" title="E108">C0</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">seqt1</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">H1</font>,<font color="Maroon" title="c5">phi1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c13">seqt2</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">H2</font>,<font color="Maroon" title="c7">phi</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as  <font color="Maroon" title="c2">S</font> <a href="fomodel4.html#V1" title="FOMODEL4:attr.1">-sequent-like</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:192"><span class="lab"><font color="Green" title="E109">C1</font></span></a>, <a class="txt" href="#E10:192"><span class="lab"><font color="Green" title="E110">C2</font></span></a></span>;<br/>
<a NAME="E12:192"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c12">seqt1</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <font color="Maroon" title="c8">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c13">seqt2</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <font color="Maroon" title="c10">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">H11</font> = <font color="Maroon" title="c8">H1</font>, <font color="Maroon" title="c15">H22</font> = <font color="Maroon" title="c10">H2</font> as  <font color="Maroon" title="c2">S</font> <a href="fomodel4.html#V13" title="FOMODEL4:attr.13">-premises-like</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:192"><span class="lab"><font color="Green" title="E109">C1</font></span></a>, <a class="txt" href="#E10:192"><span class="lab"><font color="Green" title="E110">C2</font></span></a>, <a class="ref" href="fomodel0.html#T29" title="FOMODEL0:th.29">FOMODEL0:29</a></span>;<br/>
<a NAME="E14:192"/><span class="lab"><font color="Green" title="E111">CC3</font></span>: 
<font color="Maroon" title="c5">phi1</font> is <font color="Maroon" title="c8">H1</font> <a href="fomodel0.html#K13" title="FOMODEL0:func.13">null</a> <font color="Maroon" title="c10">H2</font>,<font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:192"><span class="lab"><font color="Green" title="E109">C1</font></span></a>, <a class="ref" href="fomodel4.html#D62" target="_self" title="FOMODEL4:def.62">DefProvable3</a></span>;<br/>
<a NAME="E15:192"/>
<font color="Maroon" title="c7">phi</font> is <font color="Maroon" title="c10">H2</font> <a href="fomodel0.html#K13" title="FOMODEL0:func.13">null</a> <font color="Maroon" title="c8">H1</font>,<font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:192"><span class="lab"><font color="Green" title="E110">C2</font></span></a>, <a class="ref" href="fomodel4.html#D62" target="_self" title="FOMODEL4:def.62">DefProvable3</a></span>;<br/>
<a NAME="E16:192"/><span class="kw">then </span>
<font color="Maroon" title="c14">H11</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">H22</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="#E14:192"><span class="lab"><font color="Green" title="E111">CC3</font></span></a>, <a class="ref" href="fomodel4.html#D65" target="_self" title="FOMODEL4:def.65">DefInc</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:192"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:192"><span class="lab"><font color="Green" title="E107">C4</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/>


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