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<div class="add">

<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c2">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/>



<b>set </b><font color="Maroon" title="c3">n2</font> = <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2;<br/>
<b>set </b><font color="Maroon" title="c4">2SS</font> =  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span>;<br/>
<a NAME="E1:21"/><i><font color="Green" title="E17">A1</font></i>: 
 <a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_2.html#D17" title="MATRIX_2:def.17">MATRIX_2:def 17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">2S</font> =  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span> ;<br/>
<b>let </b><font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c7">q</font>, <font color="Maroon" title="c8">pq</font> be    <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E3:21"/><i><font color="Green" title="E18">A2</font></i>: 
( <font color="Maroon" title="c8">pq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c7">q</font> <a href="matrix_2.html#NR1" title="MATRIX_2:NR.1">is_transposition</a>  )
 ;<br/>

<a NAME="E4:21"/><i><font color="Green" title="E19">A3</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Maroon" title="c7">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font> holds <br/> <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c10">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:21_1"/><i><font color="Green" title="E20">A4</font></i>: 
( <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">j</font> &amp; <font color="Maroon" title="c7">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_1_1"/>
(  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font></span>)</span> or  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1"><b>suppose </b></a><a NAME="E1:21_1_1_1_1"/>
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:21_1_1_1_1"/><b>then </b>
(  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font></span>)</span> &amp;  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font> )
 <b>by </b><i><a class="ref" href="matrix11.html#T11" target="_self" title="MATRIX11:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E3:21_1_1_1_1"/>
 <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_2"><b>suppose </b></a><a NAME="E1:21_1_1_1_2"/><i><font color="Green" title="E21">A5</font></i>: 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font></span>)</span>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c11">P1</font> =  <a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font>;<br/><b>set </b><font color="Maroon" title="c12">P2</font> =  <a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font>;<br/><b>set </b><font color="Maroon" title="c13">D</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> : ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">2S</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> ) </span>}</span> ;<br/><a NAME="E2:21_1_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> : ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">2S</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">2S</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:21_1_1_1_2_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> : ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">2S</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:21_1_1_1_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">2S</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> )
 
;<br/>
<b>hence </b><a NAME="E3:21_1_1_1_2_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">2S</font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c14">D</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> : ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">2S</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> ) </span>}</span>  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>  <b>by </b><i><a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/><b>set </b><font color="Maroon" title="c15">D1</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> : ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">2S</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">s</font> ) </span>}</span> ;<br/><b>set </b><font color="Maroon" title="c16">D2</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> : ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">2S</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">s</font> ) </span>}</span> ;<br/><a NAME="E4:21_1_1_1_2"/><i><font color="Green" title="E22">A6</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> : ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">2S</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">s</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">D</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:21_1_1_1_2_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> : ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">2S</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">s</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:21_1_1_1_2_2"/><b>then </b>
 ex <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> st <br/>( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">2S</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">s</font> )
 
;<br/>
<b>hence </b><a NAME="E3:21_1_1_1_2_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E5:21_1_1_1_2"/><i><font color="Green" title="E23">A7</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> : ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">2S</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">s</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">D</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:21_1_1_1_2_3"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> : ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">2S</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">s</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:21_1_1_1_2_3"/><b>then </b>
 ex <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> st <br/>( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">2S</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">s</font> )
 
;<br/>
<b>hence </b><a NAME="E3:21_1_1_1_2_3"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c17">D1</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> : ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">2S</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">s</font> ) </span>}</span> , <font color="Maroon" title="c18">D2</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> : ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">2S</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">s</font> ) </span>}</span>  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>  <b>by </b><i><a class="txt" href="matrix11.html#E4:21_1_1_1_2"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/><a NAME="E7:21_1_1_1_2"/><i><font color="Green" title="E24">A8</font></i>: 
<font color="Maroon" title="c14">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">D1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">D2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:21_1_1_1_2_4"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c20">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:21_1_1_1_2_4"/><i><font color="Green" title="E25">A9</font></i>: 
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">s</font> &amp; <font color="Maroon" title="c20">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">2S</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">s</font> )
 ;<br/>
<a NAME="E4:21_1_1_1_2_4"/>
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">s</font> or <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">s</font> )
 
<b>by </b><i><a class="txt" href="matrix11.html#E3:21"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="matrix11.html#E1:21_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="matrix11.html#E3:21_1_1_1_2_4"><i><font color="Green" title="E25">A9</font></i></a>, <a class="ref" href="matrix11.html#T9" target="_self" title="MATRIX11:th.9">Th9</a></i>;<br/>
<a NAME="E5:21_1_1_1_2_4"/><b>then </b>
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D1</font> or <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">D2</font> )
 
<b>by </b><i><a class="txt" href="matrix11.html#E3:21_1_1_1_2_4"><i><font color="Green" title="E25">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:21_1_1_1_2_4"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">D2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:21_1_1_1_2"/>
<font color="Maroon" title="c7">q</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="matrix_2.html#D11" title="MATRIX_2:def.11">MATRIX_2:def 11</a></i>;<br/><a NAME="E9:21_1_1_1_2"/><b>then </b><i><font color="Green" title="E25">A10</font></i>: 
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">q</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="matrix11.html#E3:21"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="matrix11.html#E1:21_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="matrix11.html#T8" target="_self" title="MATRIX11:th.8">Th8</a>, <a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a></i>;<br/><a NAME="E10:21_1_1_1_2"/><b>then </b><i><font color="Green" title="E26">A11</font></i>: 
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">2S</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="txt" href="matrix11.html#E3:21"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="matrix11.html#E1:21_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="matrix11.html#E1:21_1_1_1_2"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="matrix11.html#T1" target="_self" title="MATRIX11:th.1">Th1</a>, <a class="ref" href="matrix11.html#T10" target="_self" title="MATRIX11:th.10">Th10</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E11:21_1_1_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D1</font> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">D2</font> )
 ;<br/><a NAME="E12:21_1_1_1_2"/><b>then </b><i><font color="Green" title="E27">A12</font></i>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">D2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E13:21_1_1_1_2"/><i><font color="Green" title="E28">A13</font></i>: 
<font color="Maroon" title="c17">D1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">D2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:21_1_1_1_2_5"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">D2</font>
 ;<br/>

<a NAME="E2:21_1_1_1_2_5"/><b>then </b><i><font color="Green" title="E29">A14</font></i>: 
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D1</font> &amp; <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">D2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">s1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:21_1_1_1_2_5"/><i><font color="Green" title="E30">A15</font></i>: 
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">s1</font> &amp; <font color="Maroon" title="c20">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">2S</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">s1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">s1</font> &amp; <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">s1</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c21">s2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span><b> such that </b><br/><a NAME="E6:21_1_1_1_2_5"/><i><font color="Green" title="E31">A16</font></i>: 
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">s2</font> &amp; <font color="Maroon" title="c21">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">2S</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s2</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">s2</font> )
 <b>by </b><i><a class="txt" href="matrix11.html#E2:21_1_1_1_2_5"><i><font color="Green" title="E29">A14</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">i'</font>, <font color="Maroon" title="c23">j'</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E8:21_1_1_1_2_5"/><i><font color="Green" title="E32">A17</font></i>: 
( <font color="Maroon" title="c22">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> &amp; <font color="Maroon" title="c23">j'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> &amp; <font color="Maroon" title="c22">i'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">j'</font> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c22">i'</font>,<font color="Maroon" title="c23">j'</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x</font> )
 <b>by </b><i><a class="txt" href="matrix11.html#E4:21_1_1_1_2_5"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="matrix11.html#T1" target="_self" title="MATRIX11:th.1">Th1</a></i>;<br/>
<a NAME="E9:21_1_1_1_2_5"/>
( ( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i'</font> or <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j'</font> ) &amp; ( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i'</font> or <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j'</font> ) )
 
<b>by </b><i><a class="txt" href="matrix11.html#E4:21_1_1_1_2_5"><i><font color="Green" title="E30">A15</font></i></a>, <a class="txt" href="matrix11.html#E6:21_1_1_1_2_5"><i><font color="Green" title="E31">A16</font></i></a>, <a class="txt" href="matrix11.html#E8:21_1_1_1_2_5"><i><font color="Green" title="E32">A17</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E10:21_1_1_1_2_5"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="matrix11.html#E1:21_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="matrix11.html#E8:21_1_1_1_2_5"><i><font color="Green" title="E32">A17</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">i</font> holds <br/>( ( <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">j</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c10">j</font>,<font color="Olive" title="b1">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) &amp; ( <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) );<br/><a NAME="E14:21_1_1_1_2"/><i><font color="Green" title="E29">A18</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D1</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">D2</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:21_1_1_1_2_6"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c20">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:21_1_1_1_2_6"/><i><font color="Green" title="E30">A19</font></i>: 
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">s</font> &amp; <font color="Maroon" title="c20">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">2S</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">s</font> &amp; <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">s</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c21">j'</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:21_1_1_1_2_6"/><i><font color="Green" title="E31">A20</font></i>: 
( <font color="Maroon" title="c21">j'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> &amp; <font color="Maroon" title="c21">j'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c20">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c21">j'</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="txt" href="matrix11.html#E3:21_1_1_1_2_6"><i><font color="Green" title="E30">A19</font></i></a>, <a class="txt" href="matrix11.html#E16"><i><font color="Green" title="E15">Lm2</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_2_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_2_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_1_1_2_6_1_1"/>
( <font color="Maroon" title="c21">j'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j</font> or <font color="Maroon" title="c21">j'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_2_6_1_1_1"><b>suppose </b></a><a NAME="E1:21_1_1_1_2_6_1_1_1"/><i><font color="Green" title="E32">A21</font></i>: 
<font color="Maroon" title="c21">j'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c22">X</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>;<br/><b>thus </b><a NAME="E2:21_1_1_1_2_6_1_1_1"/>
<font color="Maroon" title="c22">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">D2</font>
 <b>by </b><i><a class="txt" href="matrix11.html#E10:21_1_1_1_2"><i><font color="Green" title="E26">A11</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c23">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E3:21_1_1_1_2_6_1_1_1"/><i><font color="Green" title="E33">A22</font></i>: 
( <font color="Maroon" title="c23">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">x</font> &amp; <font color="Maroon" title="c23">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">i</font> )
 ;<br/><b>thus </b><a NAME="E4:21_1_1_1_2_6_1_1_1"/>
( ( <font color="Maroon" title="c23">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">j</font> implies <font color="Maroon" title="c22">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c23">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) &amp; ( <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j</font> implies <font color="Maroon" title="c22">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) )
 <b>by </b><i><a class="txt" href="matrix11.html#E3:21_1_1_1_2_6"><i><font color="Green" title="E30">A19</font></i></a>, <a class="txt" href="matrix11.html#E5:21_1_1_1_2_6"><i><font color="Green" title="E31">A20</font></i></a>, <a class="txt" href="matrix11.html#E1:21_1_1_1_2_6_1_1_1"><i><font color="Green" title="E32">A21</font></i></a>, <a class="txt" href="matrix11.html#E3:21_1_1_1_2_6_1_1_1"><i><font color="Green" title="E33">A22</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_2_6_1_1_2"><b>suppose </b></a><a NAME="E1:21_1_1_1_2_6_1_1_2"/><i><font color="Green" title="E32">A23</font></i>: 
<font color="Maroon" title="c21">j'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">j</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c22">X</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c21">j'</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>;<br/><a NAME="E2:21_1_1_1_2_6_1_1_2"/>
( <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">j'</font> or <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c21">j'</font> )
 <b>by </b><i><a class="txt" href="matrix11.html#E1:21_1_1_1_2_6_1_1_2"><i><font color="Green" title="E32">A23</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:21_1_1_1_2_6_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">X</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">X</font> &amp; <font color="Maroon" title="c22">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix11.html#E3:21"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="matrix11.html#E1:21_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="matrix11.html#E1:21_1_1_1_2"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="matrix11.html#E9:21_1_1_1_2"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="matrix11.html#E3:21_1_1_1_2_6"><i><font color="Green" title="E30">A19</font></i></a>, <a class="txt" href="matrix11.html#E5:21_1_1_1_2_6"><i><font color="Green" title="E31">A20</font></i></a>, <a class="ref" href="matrix11.html#T1" target="_self" title="MATRIX11:th.1">Th1</a>, <a class="ref" href="matrix11.html#T10" target="_self" title="MATRIX11:th.10">Th10</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E4:21_1_1_1_2_6_1_1_2"/>
<font color="Maroon" title="c22">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">D2</font>
 ;<br/><b>let </b><font color="Maroon" title="c23">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E5:21_1_1_1_2_6_1_1_2"/><i><font color="Green" title="E33">A24</font></i>: 
( <font color="Maroon" title="c23">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">x</font> &amp; <font color="Maroon" title="c23">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">i</font> )
 ;<br/><b>thus </b><a NAME="E6:21_1_1_1_2_6_1_1_2"/>
( ( <font color="Maroon" title="c23">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">j</font> implies <font color="Maroon" title="c22">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c23">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) &amp; ( <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j</font> implies <font color="Maroon" title="c22">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) )
 <b>by </b><i><a class="txt" href="matrix11.html#E3:21_1_1_1_2_6"><i><font color="Green" title="E30">A19</font></i></a>, <a class="txt" href="matrix11.html#E5:21_1_1_1_2_6"><i><font color="Green" title="E31">A20</font></i></a>, <a class="txt" href="matrix11.html#E1:21_1_1_1_2_6_1_1_2"><i><font color="Green" title="E32">A23</font></i></a>, <a class="txt" href="matrix11.html#E5:21_1_1_1_2_6_1_1_2"><i><font color="Green" title="E33">A24</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:21_1_1_1_2_6"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">D2</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c19">x</font>,<font color="Olive" title="b1">y</font>] )
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c19">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c17">D1</font>,<font color="Maroon" title="c18">D2</font><b> such that </b><br/><a NAME="E16:21_1_1_1_2"/><i><font color="Green" title="E30">A25</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="matrix11.html#E14:21_1_1_1_2"><i><font color="Green" title="E29">A18</font></i></a>);<br/></i><a NAME="E17:21_1_1_1_2"/><i><font color="Green" title="E31">A26</font></i>: 
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D1</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D1</font> &amp; <font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">x1</font>, <font color="Maroon" title="c21">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:21_1_1_1_2_7"/><i><font color="Green" title="E32">A27</font></i>: 
( <font color="Maroon" title="c20">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D1</font> &amp; <font color="Maroon" title="c21">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D1</font> &amp; <font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x2</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c22">s1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:21_1_1_1_2_7"/><i><font color="Green" title="E33">A28</font></i>: 
( <font color="Maroon" title="c20">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">s1</font> &amp; <font color="Maroon" title="c22">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">2S</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">s1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">s1</font> &amp; <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">s1</font> )
 <b>by </b><i><a class="txt" href="matrix11.html#E1:21_1_1_1_2_7"><i><font color="Green" title="E32">A27</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c23">j1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:21_1_1_1_2_7"/><i><font color="Green" title="E34">A29</font></i>: 
( <font color="Maroon" title="c23">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> &amp; <font color="Maroon" title="c9">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">j1</font> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c23">j1</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">s1</font> )
 <b>by </b><i><a class="txt" href="matrix11.html#E3:21_1_1_1_2_7"><i><font color="Green" title="E33">A28</font></i></a>, <a class="txt" href="matrix11.html#E16"><i><font color="Green" title="E15">Lm2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c24">s2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span><b> such that </b><br/><a NAME="E7:21_1_1_1_2_7"/><i><font color="Green" title="E35">A30</font></i>: 
( <font color="Maroon" title="c21">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">s2</font> &amp; <font color="Maroon" title="c24">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">2S</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">s2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">s2</font> &amp; <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">s2</font> )
 <b>by </b><i><a class="txt" href="matrix11.html#E1:21_1_1_1_2_7"><i><font color="Green" title="E32">A27</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">j2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:21_1_1_1_2_7"/><i><font color="Green" title="E36">A31</font></i>: 
( <font color="Maroon" title="c25">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> &amp; <font color="Maroon" title="c9">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">j2</font> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c25">j2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">s2</font> )
 <b>by </b><i><a class="txt" href="matrix11.html#E7:21_1_1_1_2_7"><i><font color="Green" title="E35">A30</font></i></a>, <a class="txt" href="matrix11.html#E16"><i><font color="Green" title="E15">Lm2</font></i></a></i>;<br/>
<a NAME="E10:21_1_1_1_2_7"/><i><font color="Green" title="E37">A32</font></i>: 
( <font color="Maroon" title="c23">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">s1</font> &amp; <font color="Maroon" title="c25">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">s2</font> )
 
<b>by </b><i><a class="txt" href="matrix11.html#E5:21_1_1_1_2_7"><i><font color="Green" title="E34">A29</font></i></a>, <a class="txt" href="matrix11.html#E9:21_1_1_1_2_7"><i><font color="Green" title="E36">A31</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_2_7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_2_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_1_1_2_7_1_1"/>
( ( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2</font> ) or ( <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">j1</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2</font> ) or ( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">j2</font> ) or ( <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">j1</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">j2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_2_7_1_1_1"><b>case </b></a><a NAME="E1:21_1_1_1_2_7_1_1_1"/>
( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1_1_2_7_1_1_1"/>
<font color="Maroon" title="c20">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x2</font>
 <b>by </b><i><a class="txt" href="matrix11.html#E3:21_1_1_1_2_7"><i><font color="Green" title="E33">A28</font></i></a>, <a class="txt" href="matrix11.html#E5:21_1_1_1_2_7"><i><font color="Green" title="E34">A29</font></i></a>, <a class="txt" href="matrix11.html#E7:21_1_1_1_2_7"><i><font color="Green" title="E35">A30</font></i></a>, <a class="txt" href="matrix11.html#E9:21_1_1_1_2_7"><i><font color="Green" title="E36">A31</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_2_7_1_1_2"><b>case </b></a><a NAME="E1:21_1_1_1_2_7_1_1_2"/>
( <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">j1</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j2</font> )
 ;<br/><div class="add"><a NAME="E2:21_1_1_1_2_7_1_1_2"/><b>then </b>
( <font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c23">j1</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="txt" href="matrix11.html#E16:21_1_1_1_2"><i><font color="Green" title="E30">A25</font></i></a>, <a class="txt" href="matrix11.html#E1:21_1_1_1_2_7"><i><font color="Green" title="E32">A27</font></i></a>, <a class="txt" href="matrix11.html#E3:21_1_1_1_2_7"><i><font color="Green" title="E33">A28</font></i></a>, <a class="txt" href="matrix11.html#E5:21_1_1_1_2_7"><i><font color="Green" title="E34">A29</font></i></a>, <a class="txt" href="matrix11.html#E7:21_1_1_1_2_7"><i><font color="Green" title="E35">A30</font></i></a>, <a class="txt" href="matrix11.html#E9:21_1_1_1_2_7"><i><font color="Green" title="E36">A31</font></i></a>, <a class="txt" href="matrix11.html#E10:21_1_1_1_2_7"><i><font color="Green" title="E37">A32</font></i></a></i>;<br/><b>hence </b><a NAME="E3:21_1_1_1_2_7_1_1_2"/>
<font color="Maroon" title="c20">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x2</font>
 <b>by </b><i><a class="txt" href="matrix11.html#E1:21_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="matrix11.html#E1:21_1_1_1_2_7"><i><font color="Green" title="E32">A27</font></i></a>, <a class="txt" href="matrix11.html#E5:21_1_1_1_2_7"><i><font color="Green" title="E34">A29</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_2_7_1_1_3"><b>case </b></a><a NAME="E1:21_1_1_1_2_7_1_1_3"/>
( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j1</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">j2</font> )
 ;<br/><div class="add"><a NAME="E2:21_1_1_1_2_7_1_1_3"/><b>then </b>
( <font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c25">j2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="txt" href="matrix11.html#E16:21_1_1_1_2"><i><font color="Green" title="E30">A25</font></i></a>, <a class="txt" href="matrix11.html#E1:21_1_1_1_2_7"><i><font color="Green" title="E32">A27</font></i></a>, <a class="txt" href="matrix11.html#E3:21_1_1_1_2_7"><i><font color="Green" title="E33">A28</font></i></a>, <a class="txt" href="matrix11.html#E5:21_1_1_1_2_7"><i><font color="Green" title="E34">A29</font></i></a>, <a class="txt" href="matrix11.html#E7:21_1_1_1_2_7"><i><font color="Green" title="E35">A30</font></i></a>, <a class="txt" href="matrix11.html#E9:21_1_1_1_2_7"><i><font color="Green" title="E36">A31</font></i></a>, <a class="txt" href="matrix11.html#E10:21_1_1_1_2_7"><i><font color="Green" title="E37">A32</font></i></a></i>;<br/><b>hence </b><a NAME="E3:21_1_1_1_2_7_1_1_3"/>
<font color="Maroon" title="c20">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x2</font>
 <b>by </b><i><a class="txt" href="matrix11.html#E1:21_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="matrix11.html#E1:21_1_1_1_2_7"><i><font color="Green" title="E32">A27</font></i></a>, <a class="txt" href="matrix11.html#E9:21_1_1_1_2_7"><i><font color="Green" title="E36">A31</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_2_7_1_1_4"><b>case </b></a><a NAME="E1:21_1_1_1_2_7_1_1_4"/><i><font color="Green" title="E38">A33</font></i>: 
( <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">j1</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">j2</font> )
 ;<br/><div class="add"><a NAME="E2:21_1_1_1_2_7_1_1_4"/><b>then </b>
( <font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c23">j1</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c25">j2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c23">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c23">j1</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="txt" href="matrix11.html#E16:21_1_1_1_2"><i><font color="Green" title="E30">A25</font></i></a>, <a class="txt" href="matrix11.html#E1:21_1_1_1_2_7"><i><font color="Green" title="E32">A27</font></i></a>, <a class="txt" href="matrix11.html#E3:21_1_1_1_2_7"><i><font color="Green" title="E33">A28</font></i></a>, <a class="txt" href="matrix11.html#E5:21_1_1_1_2_7"><i><font color="Green" title="E34">A29</font></i></a>, <a class="txt" href="matrix11.html#E7:21_1_1_1_2_7"><i><font color="Green" title="E35">A30</font></i></a>, <a class="txt" href="matrix11.html#E9:21_1_1_1_2_7"><i><font color="Green" title="E36">A31</font></i></a>, <a class="txt" href="matrix11.html#E10:21_1_1_1_2_7"><i><font color="Green" title="E37">A32</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:21_1_1_1_2_7_1_1_4"/>
<font color="Maroon" title="c20">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x2</font>
 <b>by </b><i><a class="txt" href="matrix11.html#E1:21_1_1_1_2_7"><i><font color="Green" title="E32">A27</font></i></a>, <a class="txt" href="matrix11.html#E3:21_1_1_1_2_7"><i><font color="Green" title="E33">A28</font></i></a>, <a class="txt" href="matrix11.html#E5:21_1_1_1_2_7"><i><font color="Green" title="E34">A29</font></i></a>, <a class="txt" href="matrix11.html#E7:21_1_1_1_2_7"><i><font color="Green" title="E35">A30</font></i></a>, <a class="txt" href="matrix11.html#E9:21_1_1_1_2_7"><i><font color="Green" title="E36">A31</font></i></a>, <a class="txt" href="matrix11.html#E1:21_1_1_1_2_7_1_1_4"><i><font color="Green" title="E38">A33</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E12:21_1_1_1_2_7"/>
<font color="Maroon" title="c20">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x2</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E18:21_1_1_1_2"/><i><font color="Green" title="E32">A34</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">D2</font> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D1</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:21_1_1_1_2_8"/>
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">D2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c21">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:21_1_1_1_2_8"/><i><font color="Green" title="E33">A35</font></i>: 
( <font color="Maroon" title="c21">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">y</font> &amp; <font color="Maroon" title="c21">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">2S</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">s</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c22">i1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:21_1_1_1_2_8"/><i><font color="Green" title="E34">A36</font></i>: 
( <font color="Maroon" title="c22">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> &amp; <font color="Maroon" title="c22">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">j</font> &amp; <font color="Maroon" title="c21">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c22">i1</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="txt" href="matrix11.html#E3:21_1_1_1_2_8"><i><font color="Green" title="E33">A35</font></i></a>, <a class="txt" href="matrix11.html#E16"><i><font color="Green" title="E15">Lm2</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_2_8_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_2_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_1_1_2_8_1_1"/>
( <font color="Maroon" title="c22">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i</font> or <font color="Maroon" title="c22">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_2_8_1_1_1"><b>suppose </b></a><a NAME="E1:21_1_1_1_2_8_1_1_1"/>
<font color="Maroon" title="c22">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i</font>
 ;<br/><div class="add"><a NAME="E2:21_1_1_1_2_8_1_1_1"/><b>then </b>
( <font color="Maroon" title="c21">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D1</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="txt" href="matrix11.html#E10:21_1_1_1_2"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="matrix11.html#E5:21_1_1_1_2_8"><i><font color="Green" title="E34">A36</font></i></a></i>;<br/><a NAME="E3:21_1_1_1_2_8_1_1_1"/><b>then </b>
( <font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">y</font> &amp; <font color="Maroon" title="c21">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D1</font> )
 <b>by </b><i><a class="txt" href="matrix11.html#E1:21_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="matrix11.html#E16:21_1_1_1_2"><i><font color="Green" title="E30">A25</font></i></a>, <a class="txt" href="matrix11.html#E3:21_1_1_1_2_8"><i><font color="Green" title="E33">A35</font></i></a></i>;<br/><b>hence </b><a NAME="E4:21_1_1_1_2_8_1_1_1"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D1</font> &amp; <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_2_8_1_1_2"><b>suppose </b></a><a NAME="E1:21_1_1_1_2_8_1_1_2"/><i><font color="Green" title="E35">A37</font></i>: 
<font color="Maroon" title="c22">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">i</font>
 ;<br/><div class="add"><a NAME="E2:21_1_1_1_2_8_1_1_2"/><b>then </b>
( <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">i1</font> or <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c22">i1</font> )
 <b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:21_1_1_1_2_8_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c22">i1</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c22">i1</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c22">i1</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c22">i1</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="txt" href="matrix11.html#E3:21"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="matrix11.html#E1:21_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="matrix11.html#E1:21_1_1_1_2"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="matrix11.html#E9:21_1_1_1_2"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="matrix11.html#E3:21_1_1_1_2_8"><i><font color="Green" title="E33">A35</font></i></a>, <a class="txt" href="matrix11.html#E5:21_1_1_1_2_8"><i><font color="Green" title="E34">A36</font></i></a>, <a class="ref" href="matrix11.html#T1" target="_self" title="MATRIX11:th.1">Th1</a>, <a class="ref" href="matrix11.html#T10" target="_self" title="MATRIX11:th.10">Th10</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E4:21_1_1_1_2_8_1_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c22">i1</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D1</font> &amp; <font color="Maroon" title="c22">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c22">i1</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E5:21_1_1_1_2_8_1_1_2"/><b>then </b>
( <font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c22">i1</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c22">i1</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c22">i1</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D1</font> )
 <b>by </b><i><a class="txt" href="matrix11.html#E16:21_1_1_1_2"><i><font color="Green" title="E30">A25</font></i></a>, <a class="txt" href="matrix11.html#E5:21_1_1_1_2_8"><i><font color="Green" title="E34">A36</font></i></a>, <a class="txt" href="matrix11.html#E1:21_1_1_1_2_8_1_1_2"><i><font color="Green" title="E35">A37</font></i></a></i>;<br/><b>hence </b><a NAME="E6:21_1_1_1_2_8_1_1_2"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D1</font> &amp; <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )
 <b>by </b><i><a class="txt" href="matrix11.html#E3:21_1_1_1_2_8"><i><font color="Green" title="E33">A35</font></i></a>, <a class="txt" href="matrix11.html#E5:21_1_1_1_2_8"><i><font color="Green" title="E34">A36</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:21_1_1_1_2_8"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D1</font> &amp; <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )
 ;<br/>


</div><b>end;</b></div><a NAME="E19:21_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D1</font>
 <b>by </b><i><a class="txt" href="matrix11.html#E10:21_1_1_1_2"><i><font color="Green" title="E26">A11</font></i></a></i>;<br/><a NAME="E20:21_1_1_1_2"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c17">D1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/><a NAME="E21:21_1_1_1_2"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c17">D1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c20">c1</font> = <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c17">D1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></i>;<br/><a NAME="E23:21_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">D2</font>
 <b>by </b><i><a class="txt" href="matrix11.html#E10:21_1_1_1_2"><i><font color="Green" title="E26">A11</font></i></a></i>;<br/><a NAME="E24:21_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c19">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">D2</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">D1</font> )
 <b>by </b><i><a class="txt" href="matrix11.html#E17:21_1_1_1_2"><i><font color="Green" title="E31">A26</font></i></a>, <a class="txt" href="matrix11.html#E18:21_1_1_1_2"><i><font color="Green" title="E32">A34</font></i></a>, <a class="ref" href="funct_2.html#T16" title="FUNCT_2:th.16">FUNCT_2:16</a>, <a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E25:21_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c17">D1</font>,<font color="Maroon" title="c18">D2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a>  &amp; <font color="Maroon" title="c17">D1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">D2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">D</font> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">D1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">D2</font> )
 <b>by </b><i><a class="txt" href="matrix11.html#E4:21_1_1_1_2"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="matrix11.html#E5:21_1_1_1_2"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="matrix11.html#E12:21_1_1_1_2"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><a NAME="E26:21_1_1_1_2"/><b>then </b>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c17">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">D2</font> &amp; <font color="Maroon" title="c17">D1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">D</font> &amp; <font color="Maroon" title="c17">D1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="matrix11.html#E7:21_1_1_1_2"><i><font color="Green" title="E24">A8</font></i></a>, <a class="txt" href="matrix11.html#E13:21_1_1_1_2"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>then </b> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c14">D</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c17">D1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c17">D1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#T64" title="CARD_2:th.64">CARD_2:64</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">c1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">c1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<b>by </b><i><a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">c1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1

;<br/><a NAME="E28:21_1_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c14">D</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 &amp; 1 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="nat_d.html#T14" title="NAT_D:th.14">NAT_D:14</a>, <a class="ref" href="nat_d.html#T21" title="NAT_D:th.21">NAT_D:21</a></i>;<br/><b>hence </b><a NAME="E29:21_1_1_1_2"/>
 <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix11.html#E1:21"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="matrix11.html#T7" target="_self" title="MATRIX11:th.7">Th7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:21_1"/>
 <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c10">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:21"/><i><font color="Green" title="E20">A38</font></i>: 
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c9">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">j</font> &amp; <font color="Maroon" title="c7">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j</font> &amp; <font color="Maroon" title="c7">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i</font> )
 <b>and </b><br/><a NAME="E7:21"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">j</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">q</font> holds <br/><font color="Maroon" title="c7">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font>
 <b>by </b><i><a class="txt" href="matrix11.html#E3:21"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="matrix_2.html#D14" title="MATRIX_2:def.14">MATRIX_2:def 14</a></i>;<br/>
<a NAME="E8:21"/>
( <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">j</font> or <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">i</font> )
 
<b>by </b><i><a class="txt" href="matrix11.html#E6:21"><i><font color="Green" title="E20">A38</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:21"/>
 <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c8">pq</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">K</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix11.html#E4:21"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="matrix11.html#E6:21"><i><font color="Green" title="E20">A38</font></i></a></i>;<br/>


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