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

<span class="kw">let </span><font color="Maroon" title="c1">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">s</font> being   <a href="fomodel1.html#V7" title="FOMODEL1:attr.7">relational</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/>  for <font color="Olive" title="b2">T1</font>, <font color="Olive" title="b3">T2</font> being   <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a>  holds  <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fomodel4.html#K36" title="FOMODEL4:func.36">PairWiseEq</a> (<font color="Olive" title="b2">T1</font>,<font color="Olive" title="b3">T2</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"><span class="p5">(<span class="default"><font color="Olive" title="b1">s</font> <a href="fomodel3.html#K1" title="FOMODEL3:func.1">-compound</a> <font color="Olive" title="b2">T1</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b1">s</font> <a href="fomodel3.html#K1" title="FOMODEL3:func.1">-compound</a> <font color="Olive" title="b3">T2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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"><a href="fomodel4.html#K28" title="FOMODEL4:func.28">R3e</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="fomodel4.html#V3" title="FOMODEL4:attr.3">-derivable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">s</font> be   <a href="fomodel1.html#V7" title="FOMODEL1:attr.7">relational</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Maroon" title="c1">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">T1</font>, <font color="Olive" title="b2">T2</font> being   <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a>  holds  <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fomodel4.html#K36" title="FOMODEL4:func.36">PairWiseEq</a> (<font color="Olive" title="b1">T1</font>,<font color="Olive" title="b2">T2</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"><span class="p5">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="fomodel3.html#K1" title="FOMODEL3:func.1">-compound</a> <font color="Olive" title="b1">T1</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="fomodel3.html#K1" title="FOMODEL3:func.1">-compound</a> <font color="Olive" title="b2">T2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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"><a href="fomodel4.html#K28" title="FOMODEL4:func.28">R3e</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="fomodel4.html#V3" title="FOMODEL4:attr.3">-derivable</a> </span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">m</font> =  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <font color="Maroon" title="c2">s</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">AT</font> =  <a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">E</font> =  <a href="fomodel2.html#K34" title="FOMODEL2:func.34">TheEqSymbOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">AF</font> =  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">T1</font>, <font color="Maroon" title="c8">T2</font> be   <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fomodel4.html#K36" title="FOMODEL4:func.36">PairWiseEq</a> (<font color="Maroon" title="c7">T1</font>,<font color="Maroon" title="c8">T2</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"><span class="p5">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="fomodel3.html#K1" title="FOMODEL3:func.1">-compound</a> <font color="Maroon" title="c7">T1</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="fomodel3.html#K1" title="FOMODEL3:func.1">-compound</a> <font color="Maroon" title="c8">T2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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"><a href="fomodel4.html#K28" title="FOMODEL4:func.28">R3e</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="fomodel4.html#V3" title="FOMODEL4:attr.3">-derivable</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">w1</font> = <font color="Maroon" title="c2">s</font> <a href="fomodel3.html#K1" title="FOMODEL3:func.1">-compound</a> <font color="Maroon" title="c7">T1</font>, <font color="Maroon" title="c10">conclusion</font> = <font color="Maroon" title="c2">s</font> <a href="fomodel3.html#K1" title="FOMODEL3:func.1">-compound</a> <font color="Maroon" title="c8">T2</font> as   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E2:103"/>
<font color="Maroon" title="c9">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c1">S</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">w11</font> = <font color="Maroon" title="c9">w1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c1">S</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">premises</font> = <span class="p1">(<span class="default"><a href="fomodel4.html#K36" title="FOMODEL4:func.36">PairWiseEq</a> (<font color="Maroon" title="c7">T1</font>,<font color="Maroon" title="c8">T2</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">w11</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c1">S</font></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">seqt</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">premises</font>,<font color="Maroon" title="c10">conclusion</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D2" target="_self" title="FOMODEL4:def.2">DefSeqtLike</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">Seqts</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>;<br/>
<a NAME="E7:103"/>
( <font color="Maroon" title="c13">seqt</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">s</font> <a href="fomodel3.html#K1" title="FOMODEL3:func.1">-compound</a> <font color="Maroon" title="c7">T1</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="fomodel4.html#K36" title="FOMODEL4:func.36">PairWiseEq</a> (<font color="Maroon" title="c7">T1</font>,<font color="Maroon" title="c8">T2</font>)</span>)</span> &amp; <font color="Maroon" title="c13">seqt</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="fomodel3.html#K1" title="FOMODEL3:func.1">-compound</a> <font color="Maroon" title="c8">T2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E8:103"/><span class="kw">then </span>
<font color="Maroon" title="c13">seqt</font> <a href="fomodel4.html#R7" title="FOMODEL4:pred.7">Rule3e</a> <font color="Maroon" title="c14">Seqts</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D24" target="_self" title="FOMODEL4:def.24">Def3e</a></span>;<br/>
<a NAME="E9:103"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">Seqts</font>,<font color="Maroon" title="c13">seqt</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel4.html#K14" title="FOMODEL4:func.14">P3e</a> <font color="Maroon" title="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D37" target="_self" title="FOMODEL4:def.37">DefP3e</a></span>;<br/>
<a NAME="E10:103"/><span class="kw">then </span>
<font color="Maroon" title="c13">seqt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fomodel4.html#K28" title="FOMODEL4:func.28">R3e</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">Seqts</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel4.html#E70"><span class="lab"><font color="Green" title="E65">Lm1</font></span></a></span>;<br/>
<a NAME="E11:103"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">seqt</font></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> <span class="p1">(<span class="default"><a href="fomodel4.html#K28" title="FOMODEL4:func.28">R3e</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">Seqts</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/>
<span class="kw">hence </span><a NAME="E12:103"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fomodel4.html#K36" title="FOMODEL4:func.36">PairWiseEq</a> (<font color="Maroon" title="c7">T1</font>,<font color="Maroon" title="c8">T2</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"><span class="p5">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="fomodel3.html#K1" title="FOMODEL3:func.1">-compound</a> <font color="Maroon" title="c7">T1</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="fomodel3.html#K1" title="FOMODEL3:func.1">-compound</a> <font color="Maroon" title="c8">T2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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"><a href="fomodel4.html#K28" title="FOMODEL4:func.28">R3e</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="fomodel4.html#V3" title="FOMODEL4:attr.3">-derivable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel4.html#E36"><span class="lab"><font color="Green" title="E31">Lm9</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>
