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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">i</font>, <font color="Olive" title="b3">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">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; <font color="Olive" title="b3">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> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">j</font> holds <br/> for <font color="Olive" title="b4">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>  st  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Olive" title="b4">M</font>,<font color="Olive" title="b2">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Olive" title="b4">M</font>,<font color="Olive" title="b3">j</font>) holds <br/> for <font color="Olive" title="b5">p</font>, <font color="Olive" title="b6">q</font>, <font color="Olive" title="b7">tr</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="b6">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b5">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b7">tr</font> &amp; <font color="Olive" title="b7">tr</font> is  <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a>  &amp; <font color="Olive" title="b7">tr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">j</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Olive" title="b4">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">q</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"><span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Olive" title="b4">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">p</font></span>)</span></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="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; <font color="Olive" title="b2">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> <font color="Maroon" title="c1">n</font> &amp; <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> holds <br/> for <font color="Olive" title="b3">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>  st  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Olive" title="b3">M</font>,<font color="Olive" title="b1">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Olive" title="b3">M</font>,<font color="Olive" title="b2">j</font>) holds <br/> for <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font>, <font color="Olive" title="b6">tr</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="b5">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b4">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b6">tr</font> &amp; <font color="Olive" title="b6">tr</font> is  <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a>  &amp; <font color="Olive" title="b6">tr</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/><span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Olive" title="b3">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">q</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"><span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Olive" title="b3">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">p</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">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; <font color="Maroon" title="c4">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> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j</font> implies  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>  st  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Olive" title="b1">M</font>,<font color="Maroon" title="c3">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Olive" title="b1">M</font>,<font color="Maroon" title="c4">j</font>) holds <br/> for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font>, <font color="Olive" title="b4">tr</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">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b4">tr</font> &amp; <font color="Olive" title="b4">tr</font> is  <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a>  &amp; <font color="Olive" title="b4">tr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Olive" title="b1">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">q</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"><span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Olive" title="b1">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font></span>)</span> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:67"/><span class="lab"><font color="Green" title="E58">A1</font></span>: 
<font color="Maroon" title="c3">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>
 <span class="kw">and </span><br/><a NAME="E2:67"/><span class="lab"><font color="Green" title="E59">A2</font></span>: 
<font color="Maroon" title="c4">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> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E3:67"/><span class="lab"><font color="Green" title="E60">A3</font></span>: 
<font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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>  st  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Olive" title="b1">M</font>,<font color="Maroon" title="c3">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Olive" title="b1">M</font>,<font color="Maroon" title="c4">j</font>) holds <br/> for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font>, <font color="Olive" title="b4">tr</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">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b4">tr</font> &amp; <font color="Olive" title="b4">tr</font> is  <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a>  &amp; <font color="Olive" title="b4">tr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Olive" title="b1">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">q</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"><span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Olive" title="b1">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font></span>)</span></span><br/>

<a NAME="E4:67"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">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>  <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> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="ref" href="matrix11.html#T1" target="_self" title="MATRIX11:th.1">Th1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">n2</font> = <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix11.html#T2" target="_self" title="MATRIX11:th.2">Th2</a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a>, <a class="ref" href="nat_1.html#T23" title="NAT_1:th.23">NAT_1:23</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">j</font>) implies  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">tr</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">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">tr</font> &amp; <font color="Olive" title="b3">tr</font> is  <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a>  &amp; <font color="Olive" title="b3">tr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c6">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">q</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"><span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c6">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E6:67"/><span class="lab"><font color="Green" title="E61">A4</font></span>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">j</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">tr</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">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">tr</font> &amp; <font color="Olive" title="b3">tr</font> is  <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a>  &amp; <font color="Olive" title="b3">tr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c6">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">q</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"><span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c6">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">M9</font> = <font color="Maroon" title="c6">M</font> as   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c5">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2,<font color="Maroon" title="c2">K</font> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">p</font>, <font color="Maroon" title="c9">q</font>, <font color="Maroon" title="c10">tr</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">tr</font> &amp; <font color="Maroon" title="c10">tr</font> is  <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a>  &amp; <font color="Maroon" title="c10">tr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> implies <span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c6">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">q</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"><span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c6">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E8:67"/><span class="lab"><font color="Green" title="E62">A5</font></span>: 
<font color="Maroon" title="c9">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">tr</font>
 <span class="kw">and </span><br/><a NAME="E9:67"/><span class="lab"><font color="Green" title="E63">A6</font></span>: 
<font color="Maroon" title="c10">tr</font> is  <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a> 
 <span class="kw">and </span><br/><a NAME="E10:67"/><span class="lab"><font color="Green" title="E64">A7</font></span>: 
<font color="Maroon" title="c10">tr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c6">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">q</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"><span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c6">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">TR</font> = <font color="Maroon" title="c10">tr</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <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="c5">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#D9" title="MATRIX_2:def.9">MATRIX_2:def 9</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">Mt</font> = <font color="Maroon" title="c7">M9</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c11">TR</font>;<br/>
<a NAME="E12:67"/><span class="lab"><font color="Green" title="E65">A8</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M9</font> holds <br/><font color="Maroon" title="c7">M9</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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">M9</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c11">TR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font> &amp; <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M9</font> implies <font color="Maroon" title="c7">M9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">M9</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c11">TR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:67_1"/><span class="lab"><font color="Green" title="E66">A9</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font>
 <span class="kw">and </span><br/><a NAME="E2:67_1"/><span class="lab"><font color="Green" title="E67">A10</font></span>: 
<font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">M9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">M9</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c11">TR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span><br/>

<a NAME="E3:67_1"/>
<font color="Maroon" title="c13">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E4:67_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A11</font></span>: 
<font color="Maroon" title="c13">k</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1"><span class="lab"><font color="Green" title="E66">A9</font></span></a>, <a class="txt" href="#E2:67_1"><span class="lab"><font color="Green" title="E67">A10</font></span></a></span>;<br/>
<a NAME="E5:67_1"/><span class="lab"><font color="Green" title="E69">A12</font></span>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="matrix_2.html#T8" title="MATRIX_2:th.8">MATRIX_2:8</a></span>;<br/>
<a NAME="E6:67_1"/><span class="lab"><font color="Green" title="E70">A13</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">TR</font> <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T52" title="FUNCT_2:th.52">FUNCT_2:52</a></span>;<br/>
<a NAME="E7:67_1"/><span class="lab"><font color="Green" title="E71">A14</font></span>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="matrix_2.html#T8" title="MATRIX_2:th.8">MATRIX_2:8</a></span>;<br/>
<a NAME="E8:67_1"/><span class="lab"><font color="Green" title="E72">A15</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D2" title="MATRIX_1:def.2">MATRIX_1:def 2</a></span>;<br/>
<a NAME="E9:67_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A16</font></span>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">M9</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c11">TR</font></span>)</span>,<font color="Maroon" title="c13">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">tr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_1"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="ref" href="matrix11.html#T38" target="_self" title="MATRIX11:th.38">Th38</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:67_1_1"/>
( <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> or <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> or ( <font color="Maroon" title="c13">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c13">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">j</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:67_1_1_1"/>
<font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">M9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">M9</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c11">TR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:67_1_1_1"/>
<font color="Maroon" title="c7">M9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">M9</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c11">TR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E6:67"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="txt" href="#E10:67"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="#E9:67_1"><span class="lab"><font color="Green" title="E73">A16</font></span></a>, <a class="txt" href="#E7:67_1"><span class="lab"><font color="Green" title="E71">A14</font></span></a>, <a class="txt" href="#E5:67_1"><span class="lab"><font color="Green" title="E69">A12</font></span></a>, <a class="ref" href="matrix_2.html#T8" title="MATRIX_2:th.8">MATRIX_2:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:67_1_1_2"/><span class="lab"><font color="Green" title="E74">A17</font></span>: 
<font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">M9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">M9</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c11">TR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span><br/><div class="add"><a NAME="E2:67_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A18</font></span>: 
<font color="Maroon" title="c6">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E6:67"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="txt" href="#E7:67_1"><span class="lab"><font color="Green" title="E71">A14</font></span></a>, <a class="ref" href="matrix_2.html#T8" title="MATRIX_2:th.8">MATRIX_2:8</a></span>;<br/><a NAME="E3:67_1_1_2"/>
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">M9</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c11">TR</font></span>)</span>,<font color="Maroon" title="c13">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="txt" href="#E9:67"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="txt" href="#E10:67"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="#E9:67_1"><span class="lab"><font color="Green" title="E73">A16</font></span></a>, <a class="txt" href="#E1:67_1_1_2"><span class="lab"><font color="Green" title="E74">A17</font></span></a>, <a class="ref" href="matrix11.html#T8" target="_self" title="MATRIX11:th.8">Th8</a></span>;<br/><span class="kw">hence </span><a NAME="E4:67_1_1_2"/>
<font color="Maroon" title="c7">M9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">M9</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c11">TR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E1:67_1_1_2"><span class="lab"><font color="Green" title="E74">A17</font></span></a>, <a class="txt" href="#E2:67_1_1_2"><span class="lab"><font color="Green" title="E75">A18</font></span></a>, <a class="ref" href="matrix_2.html#T8" title="MATRIX_2:th.8">MATRIX_2:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:67_1_1_3"/>
( <font color="Maroon" title="c13">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c13">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">j</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">M9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">M9</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c11">TR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span><br/><div class="add"><a NAME="E2:67_1_1_3"/><span class="kw">then </span>
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">M9</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c11">TR</font></span>)</span>,<font color="Maroon" title="c13">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="txt" href="#E9:67"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="txt" href="#E10:67"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="#E4:67_1"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="txt" href="#E8:67_1"><span class="lab"><font color="Green" title="E72">A15</font></span></a>, <a class="txt" href="#E6:67_1"><span class="lab"><font color="Green" title="E70">A13</font></span></a>, <a class="txt" href="#E9:67_1"><span class="lab"><font color="Green" title="E73">A16</font></span></a>, <a class="ref" href="matrix11.html#T8" target="_self" title="MATRIX11:th.8">Th8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:67_1_1_3"/>
<font color="Maroon" title="c7">M9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">M9</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c11">TR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_1"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="txt" href="#E8:67_1"><span class="lab"><font color="Green" title="E72">A15</font></span></a>, <a class="ref" href="matrix_2.html#T8" title="MATRIX_2:th.8">MATRIX_2:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E13:67"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">M9</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c11">TR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix11.html#D4" target="_self" title="MATRIX11:def.4">Def4</a></span>;<br/>
<a NAME="E14:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A19</font></span>: 
<font color="Maroon" title="c7">M9</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c11">TR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:67"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">Tr</font> = <font color="Maroon" title="c10">tr</font>, <font color="Maroon" title="c14">p2</font> = <font color="Maroon" title="c8">p</font> as    <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="c5">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> ;<br/>
<a NAME="E16:67"/><span class="lab"><font color="Green" title="E67">A20</font></span>: 
 <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> (<font color="Maroon" title="c13">Tr</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:67"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="ref" href="matrix11.html#T14" target="_self" title="MATRIX11:th.14">Th14</a></span>;<br/>
<a NAME="E17:67"/>
<font color="Maroon" title="c10">tr</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c10">tr</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:67"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="ref" href="matrix11.html#T20" target="_self" title="MATRIX11:th.20">Th20</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c6">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">q</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<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></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c7">M9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">p2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:67"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="txt" href="#E14:67"><span class="lab"><font color="Green" title="E66">A19</font></span></a>, <a class="txt" href="#E16:67"><span class="lab"><font color="Green" title="E67">A20</font></span></a>, <a class="ref" href="matrix11.html#T43" target="_self" title="MATRIX11:th.43">Th43</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c7">M9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">p2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T9" title="VECTSP_1:th.9">VECTSP_1:9</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c6">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D4" title="VECTSP_1:def.4">VECTSP_1:def 4</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
