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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">M</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">K</font><br/> for <font color="Olive" title="b3">I</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> <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <font color="Maroon" title="c1">n</font> holds <br/> <a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Olive" title="b2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Olive" title="b3">I</font>,<font color="Olive" title="b2">M</font></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">M</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font><br/> for <font color="Olive" title="b2">I</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> <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <font color="Maroon" title="c1">n</font> holds <br/> <a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Olive" title="b2">I</font>,<font color="Olive" title="b1">M</font></span><br/><b>let </b><font color="Maroon" title="c3">M</font> be   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>; <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">I</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> <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <font color="Maroon" title="c1">n</font> holds <br/> <a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Olive" title="b1">I</font>,<font color="Maroon" title="c3">M</font></span><br/><b>let </b><font color="Maroon" title="c4">I</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> <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <font color="Maroon" title="c1">n</font> implies  <a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c3">M</font> )</span><br/>







<b>assume </b><a NAME="E1:6"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c3">M</font></span><br/>

<b>set </b><font color="Maroon" title="c5">D</font> =  <a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c3">M</font>;<br/>
<b>set </b><font color="Maroon" title="c6">P</font> =  <a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c3">M</font>;<br/>
<a NAME="E2:6"/><i><font color="Green" title="E8">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c3">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_3.html#D7" title="MATRIX_3:def.7">MATRIX_3:def 7</a>, <a class="ref" href="matrix_3.html#D10" title="MATRIX_3:def.10">MATRIX_3:def 10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c3">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )</span><br/><b>assume </b><a NAME="E1:6_1"/><i><font color="Green" title="E9">A3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c3">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span><br/><a NAME="E2:6_1"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:6_1"/><b>then </b><i><font color="Green" title="E10">A4</font></i>: 
( <font color="Maroon" title="c7">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> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c3">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="txt" href="#E2:6"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="#E1:6_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:6_1"/><b>then </b>
( <font color="Maroon" title="c7">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"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c3">M</font></span>)</span> &amp; <font color="Maroon" title="c4">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i</font> )
 <b>by </b><i><a class="txt" href="#E1:6"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="finseq_2.html#T57" title="FINSEQ_2:th.57">FINSEQ_2:57</a></i>;<br/><a NAME="E5:6_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c3">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c7">i</font> &amp; <span class="p1">(<span class="default"><a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c7">i</font> )
 <b>by </b><i><a class="txt" href="#E3:6_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="matrix_3.html#D7" title="MATRIX_3:def.7">MATRIX_3:def 7</a>, <a class="ref" href="matrix_3.html#D10" title="MATRIX_3:def.10">MATRIX_3:def 10</a></i>;<br/><b>hence </b><a NAME="E6:6_1"/>
<span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c3">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 ; <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>
<b>hence </b><a NAME="E4:6"/>
 <a href="matrix_3.html#K13" title="MATRIX_3:func.13">diagonal_of_Matrix</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="txt" href="#E2:6"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>; <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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