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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">K</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/> for <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">q2</font>, <font color="Olive" title="b4">pq2</font> being   <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">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>  st <font color="Olive" title="b4">pq2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">p2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">q2</font> holds <br/> <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Olive" title="b4">pq2</font>,<font color="Olive" title="b1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Olive" title="b2">p2</font>,<font color="Olive" title="b1">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Olive" title="b3">q2</font>,<font color="Olive" title="b1">K</font></span>)</span></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p2</font>, <font color="Olive" title="b2">q2</font>, <font color="Olive" title="b3">pq2</font> being   <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">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>  st <font color="Olive" title="b3">pq2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">p2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">q2</font> holds <br/> <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Olive" title="b3">pq2</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Olive" title="b1">p2</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Olive" title="b2">q2</font>,<font color="Maroon" title="c2">K</font></span>)</span></span><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>let </b><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</font>, <font color="Maroon" title="c6">pq</font> be    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">pq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">q</font> implies  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c6">pq</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">K</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:32"/><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c6">pq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c6">pq</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">K</font></span>)</span></span><br/>

<b>consider </b><font color="Maroon" title="c7">P1</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</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="E3:32"/><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c7">P1</font>
 <b>and </b><br/><a NAME="E4:32"/><i><font color="Green" title="E30">A3</font></i>: 
for <font color="Olive" title="b1">i</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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">P1</font> holds <br/> ex <font color="Olive" title="b2">trans</font> being   <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">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> st <br/>( <font color="Maroon" title="c7">P1</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">trans</font> &amp; <font color="Olive" title="b2">trans</font> is <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a> )
 <b>by </b><i><a class="ref" href="matrix11.html#T21" target="_self" title="MATRIX11:th.21">Th21</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">P2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</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="E6:32"/><i><font color="Green" title="E31">A4</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c8">P2</font>
 <b>and </b><br/><a NAME="E7:32"/><i><font color="Green" title="E32">A5</font></i>: 
for <font color="Olive" title="b1">i</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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">P2</font> holds <br/> ex <font color="Olive" title="b2">trans</font> being   <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">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> st <br/>( <font color="Maroon" title="c8">P2</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">trans</font> &amp; <font color="Olive" title="b2">trans</font> is <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a> )
 <b>by </b><i><a class="ref" href="matrix11.html#T21" target="_self" title="MATRIX11:th.21">Th21</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">PP</font> = <font color="Maroon" title="c8">P2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">P1</font>;<br/>
<i><font color="Green" title="E33">A6</font></i>:  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">P1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c8">P2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c7">P1</font></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T8" title="GROUP_4:th.8">GROUP_4:8</a></i>
<br/>.= 
<font color="Maroon" title="c6">pq</font>
<b>by </b><i><a class="txt" href="#E1:32"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="#E3:32"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="#E6:32"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="matrix_2.html#D13" title="MATRIX_2:def.13">MATRIX_2:def 13</a></i>
;<br/>
<a NAME="E9:32"/><i><font color="Green" title="E34">A7</font></i>: 
for <font color="Olive" title="b1">i</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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">P1</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">trans</font> being   <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">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> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c8">P2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">P1</font></span>)</span> <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">trans</font> &amp; <font color="Olive" title="b2">trans</font> is <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">P1</font></span>)</span> implies  ex <font color="Olive" title="b1">trans</font> being   <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">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> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c8">P2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">trans</font> &amp; <font color="Olive" title="b1">trans</font> is <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a> ) )</span><br/>

<b>assume </b><a NAME="E1:32_2"/><i><font color="Green" title="E35">A8</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">P1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">trans</font> being   <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">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> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c8">P2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">trans</font> &amp; <font color="Olive" title="b1">trans</font> is <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_2_1_1"/>
( <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">P2</font> or  ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">P1</font> &amp; <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</font> ) )
 </span><b>by </b><i><a class="txt" href="#E1:32_2"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1_1_1"><b>suppose </b></a><a NAME="E1:32_2_1_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">P2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">trans</font> being   <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">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> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c8">P2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">trans</font> &amp; <font color="Olive" title="b1">trans</font> is <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a> )</span><br/><div class="add"><a NAME="E2:32_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c8">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> &amp;  ex <font color="Olive" title="b1">trans</font> being   <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">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> st <br/>( <font color="Maroon" title="c8">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">trans</font> &amp; <font color="Olive" title="b1">trans</font> is <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a> ) )
 <b>by </b><i><a class="txt" href="#E7:32"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E3:32_2_1_1_1"/>
 ex <font color="Olive" title="b1">trans</font> being   <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">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> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c8">P2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">trans</font> &amp; <font color="Olive" title="b1">trans</font> is <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1_1_2"><b>suppose </b></a><a NAME="E1:32_2_1_1_2"/>
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">P1</font> &amp; <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">trans</font> being   <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">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> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c8">P2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">trans</font> &amp; <font color="Olive" title="b1">trans</font> is <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a> )</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c11">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:32_2_1_1_2"/><i><font color="Green" title="E36">A9</font></i>: 
( <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">P1</font> &amp; <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">k</font> )
 ;<br/><a NAME="E4:32_2_1_1_2"/>
( <font color="Maroon" title="c7">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> &amp;  ex <font color="Olive" title="b1">trans</font> being   <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">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> st <br/>( <font color="Maroon" title="c7">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">trans</font> &amp; <font color="Olive" title="b1">trans</font> is <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a> ) )
 <b>by </b><i><a class="txt" href="#E4:32"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="#E3:32_2_1_1_2"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E5:32_2_1_1_2"/>
 ex <font color="Olive" title="b1">trans</font> being   <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">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> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c8">P2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">trans</font> &amp; <font color="Olive" title="b1">trans</font> is <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:32_2"/>
 ex <font color="Olive" title="b1">trans</font> being   <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">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> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c8">P2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">P1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">trans</font> &amp; <font color="Olive" title="b1">trans</font> is <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1"/>
( ( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">P1</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>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P2</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>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  ) or ( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">P1</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 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P2</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>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  ) or ( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">P1</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>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P2</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 ) or ( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">P1</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 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P2</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 ) )
 </span><b>by </b><i><a class="ref" href="nat_d.html#T12" title="NAT_D:th.12">NAT_D:12</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1"/><i><font color="Green" title="E35">A10</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">P1</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>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P2</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>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c6">pq</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">K</font></span>)</span></span><br/><div class="add"><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">P2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">P1</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">P1</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">P1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2
<b>by </b><i><a class="txt" href="#E1:32_3_1_1"><i><font color="Green" title="E35">A10</font></i></a>, <a class="ref" href="nat_d.html#T22" title="NAT_D:th.22">NAT_D:22</a></i>
<br/>.= 
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
<b>by </b><i><a class="txt" href="#E1:32_3_1_1"><i><font color="Green" title="E35">A10</font></i></a></i>
;<br/><a NAME="E3:32_3_1_1"/><b>then </b>
(  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c6">pq</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font> &amp;  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font> &amp;  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font> )
 <b>by </b><i><a class="txt" href="#E3:32"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="#E4:32"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="#E6:32"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="#E7:32"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="#E8:32"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="#E9:32"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="#E1:32_3_1_1"><i><font color="Green" title="E35">A10</font></i></a>, <a class="ref" href="matrix11.html#T15" target="_self" title="MATRIX11:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E4:32_3_1_1"/>
 <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c6">pq</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">K</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_2"/><i><font color="Green" title="E35">A11</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">P1</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 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P2</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>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c6">pq</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">K</font></span>)</span></span><br/><div class="add"><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">P2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">P1</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">P1</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">P1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2
<b>by </b><i><a class="txt" href="#E1:32_3_1_2"><i><font color="Green" title="E35">A11</font></i></a>, <a class="ref" href="nat_d.html#T22" title="NAT_D:th.22">NAT_D:22</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="#E1:32_3_1_2"><i><font color="Green" title="E35">A11</font></i></a></i>
;<br/><a NAME="E3:32_3_1_2"/><b>then </b>
(  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c6">pq</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">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="c4">p</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">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="c5">q</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font> )
 <b>by </b><i><a class="txt" href="#E3:32"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="#E4:32"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="#E6:32"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="#E7:32"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="#E8:32"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="#E9:32"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="#E1:32_3_1_2"><i><font color="Green" title="E35">A11</font></i></a>, <a class="ref" href="matrix11.html#T15" target="_self" title="MATRIX11:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E4:32_3_1_2"/>
 <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c6">pq</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">K</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_3"><b>suppose </b></a><a NAME="E1:32_3_1_3"/><i><font color="Green" title="E35">A12</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">P1</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>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P2</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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c6">pq</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">K</font></span>)</span></span><br/><div class="add"><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">P2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">P1</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">P1</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">P1</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2
<b>by </b><i><a class="txt" href="#E1:32_3_1_3"><i><font color="Green" title="E35">A12</font></i></a>, <a class="ref" href="nat_d.html#T22" title="NAT_D:th.22">NAT_D:22</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2
<b>by </b><i><a class="txt" href="#E1:32_3_1_3"><i><font color="Green" title="E35">A12</font></i></a>, <a class="ref" href="nat_d.html#T22" title="NAT_D:th.22">NAT_D:22</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="nat_d.html#T14" title="NAT_D:th.14">NAT_D:14</a></i>
;<br/><a NAME="E3:32_3_1_3"/><b>then </b>
(  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c6">pq</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">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="c4">p</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font> &amp;  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E3:32"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="#E4:32"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="#E6:32"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="#E7:32"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="#E8:32"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="#E9:32"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="#E1:32_3_1_3"><i><font color="Green" title="E35">A12</font></i></a>, <a class="ref" href="matrix11.html#T15" target="_self" title="MATRIX11:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E4:32_3_1_3"/>
 <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c6">pq</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">K</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_4"><b>suppose </b></a><a NAME="E1:32_3_1_4"/><i><font color="Green" title="E35">A13</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">P1</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 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P2</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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c6">pq</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">K</font></span>)</span></span><br/><div class="add"><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">P2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">P1</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">P1</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">P1</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2
<b>by </b><i><a class="txt" href="#E1:32_3_1_4"><i><font color="Green" title="E35">A13</font></i></a>, <a class="ref" href="nat_d.html#T22" title="NAT_D:th.22">NAT_D:22</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2
<b>by </b><i><a class="txt" href="#E1:32_3_1_4"><i><font color="Green" title="E35">A13</font></i></a>, <a class="ref" href="nat_d.html#T22" title="NAT_D:th.22">NAT_D:22</a></i>
<br/>.= 
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
<b>by </b><i><a class="ref" href="nat_d.html#T25" title="NAT_D:th.25">NAT_D:25</a></i>
;<br/><a NAME="E3:32_3_1_4"/><b>then </b>
(  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c6">pq</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font> &amp;  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">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="c5">q</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font> )
 <b>by </b><i><a class="txt" href="#E3:32"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="#E4:32"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="#E6:32"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="#E7:32"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="#E8:32"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="#E9:32"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="#E1:32_3_1_4"><i><font color="Green" title="E35">A13</font></i></a>, <a class="ref" href="matrix11.html#T15" target="_self" title="MATRIX11:th.15">Th15</a>, <a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>;<br/><b>hence </b><a NAME="E4:32_3_1_4"/>
 <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c6">pq</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">K</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T42" title="VECTSP_1:th.42">VECTSP_1:42</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:32"/>
 <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c6">pq</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">K</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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