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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">ITP</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><br/> for <font color="Olive" title="b2">ITG</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b2">ITG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">ITP</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/><font color="Olive" title="b1">ITP</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">ITG</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span><br/><b>let </b><font color="Maroon" title="c2">ITP</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"> for <font color="Olive" title="b1">ITG</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">ITG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">ITP</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/><font color="Maroon" title="c2">ITP</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">ITG</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span><br/><b>let </b><font color="Maroon" title="c3">ITG</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <font color="Maroon" title="c1">n</font></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="c3">ITG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">ITP</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 implies <font color="Maroon" title="c2">ITP</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">ITG</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  )</span><br/>





<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E26">A1</font></i>: 
( <font color="Maroon" title="c3">ITG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">ITP</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">ITP</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">ITG</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span><br/>

<a NAME="E2:31"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c2">ITP</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix_2.html#D11" title="MATRIX_2:def.11">MATRIX_2:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">qf</font> = <font color="Maroon" title="c2">ITP</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>, <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="matrix_2.html#D11" title="MATRIX_2:def.11">MATRIX_2:def 11</a></i>;<br/>
<a NAME="E4:31"/>
1 <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>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E1:31"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/>
<a NAME="E5:31"/><b>then </b><i><font color="Green" title="E28">A3</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">qf</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>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:31"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">qf</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>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E2:31"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E7:31"/><b>then </b><i><font color="Green" title="E29">A4</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">ITP</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c2">ITP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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> &amp; <font color="Maroon" title="c2">ITP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">ITP</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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> )
 
<b>by </b><i><a class="txt" href="matrix_7.html#E5:31"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="funct_1.html#T61" title="FUNCT_1:th.61">FUNCT_1:61</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">pf</font> = <font color="Maroon" title="c2">ITP</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="ref" href="matrix_2.html#D13" title="MATRIX_2:def.13">MATRIX_2:def 13</a></i>;<br/>
<a NAME="E9:31"/><i><font color="Green" title="E30">A5</font></i>: 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">ITG</font>,<font color="Maroon" title="c5">pf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">ITP</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c2">ITP</font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E1:31"><i><font color="Green" title="E26">A1</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="E10:31"/><i><font color="Green" title="E31">A6</font></i>: 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">pf</font>,<font color="Maroon" title="c3">ITG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">ITP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">ITP</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E1:31"><i><font color="Green" title="E26">A1</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="E11:31"/><i><font color="Green" title="E32">A7</font></i>: 
 <a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_2.html#T28" title="MATRIX_2:th.28">MATRIX_2:28</a></i>;<br/>
<a NAME="E12:31"/><i><font color="Green" title="E33">A8</font></i>: 
<font color="Maroon" title="c3">ITG</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">pf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">ITG</font>,<font color="Maroon" title="c5">pf</font>
 
;<br/>
<a NAME="E13:31"/>
<font color="Maroon" title="c5">pf</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">ITG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">pf</font>,<font color="Maroon" title="c3">ITG</font>
 
;<br/>
<b>hence </b><a NAME="E14:31"/>
<font color="Maroon" title="c2">ITP</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">ITG</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 <b>by </b><i><a class="txt" href="matrix_7.html#E7:31"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="matrix_7.html#E9:31"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="matrix_7.html#E10:31"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="matrix_7.html#E11:31"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="matrix_7.html#E12:31"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</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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