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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" title="CATALG_1:func.3">CatSign</a> <font color="Maroon" title="c1">A</font></span>)</span>  st ( <font color="Olive" title="b1">o</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">o</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) holds <br/> ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> st <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" title="CATALG_1:func.3">CatSign</a> <font color="Maroon" title="c1">A</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">o</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">o</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) implies  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> st <font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>) )</span><br/>



<span class="kw">assume </span><a NAME="E1:54"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
( <font color="Maroon" title="c2">o</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">o</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> st <font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>)</span><br/>

<a NAME="E2:54"/>
 the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" title="CATALG_1:func.3">CatSign</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="catalg_1.html#D3" target="_self" onmouseover="rs('catalg_1/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<a NAME="E3:54"/><span class="kw">then </span>
( <font color="Maroon" title="c2">o</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> or <font color="Maroon" title="c2">o</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2">(<span class="default">3 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E4:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A2</font></span>: 
( ( <font color="Maroon" title="c2">o</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c2">o</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c2">o</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c2">o</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 3 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">o</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">o</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T10" onmouseover="rs('mcart_1/T10')" onmouseout="rh()">MCART_1:10</a>, <a class="ref" href="mcart_1.html#T21" onmouseover="rs('mcart_1/T21')" onmouseout="rh()">MCART_1:21</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:54"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 <span class="kw">and </span><br/><a NAME="E7:54"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<font color="Maroon" title="c2">o</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a>, <a class="ref" href="finseq_2.html#T139" onmouseover="rs('finseq_2/T139')" onmouseout="rh()">FINSEQ_2:139</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c8">c</font> = <font color="Maroon" title="c5">c</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54"><span class="lab"><font color="Green" title="E23">A3</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">a</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> st <font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> (<font color="Maroon" title="c6">a</font>,<font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">c</font>)</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c7">b</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> st <font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Olive" title="b1">c</font>)</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c8">c</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>)</span><br/>

<span class="kw">thus </span><a NAME="E9:54"/>
<font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E4:54"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E7:54"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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