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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">  <a href="cayley.html#K2" title="CAYLEY:func.2">SymGroup</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <font color="Maroon" title="c1">n</font></span><br/>

<a NAME="E1:16"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="cayley.html#K2" title="CAYLEY:func.2">SymGroup</a> <span class="p2">(<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>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cayley.html#K1" title="CAYLEY:func.1">permutations</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="ref" href="cayley.html#D2" target="_self" title="CAYLEY:def.2">Def2</a></span>;<br/>
<span class="lab"><font color="Green" title="E9">A2</font></span>:  <a href="cayley.html#K1" title="CAYLEY:func.1">permutations</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> = 
 <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cayley.html#T3" target="_self" title="CAYLEY:th.3">Th3</a></span>
<br/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#D10" title="MATRIX_2:def.10">MATRIX_2:def 10</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="cayley.html#K2" title="CAYLEY:func.2">SymGroup</a> <span class="p2">(<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>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><a href="cayley.html#K2" title="CAYLEY:func.2">SymGroup</a> <span class="p2">(<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>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span><br/><a NAME="E1:16_2"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cayley.html#D2" target="_self" title="CAYLEY:def.2">Def2</a></span>;<br/><a NAME="E2:16_2"/>
( <font color="Maroon" title="c2">a</font> is   <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> <font color="Maroon" title="c1">n</font></span>)</span> &amp; <font color="Maroon" title="c3">b</font> is   <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> <font color="Maroon" title="c1">n</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cayley.html#T5" target="_self" title="CAYLEY:th.5">Th5</a></span>;<br/><a NAME="E3:16_2"/><span class="kw">then </span>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> )
 <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">hence </span><a NAME="E4:16_2"/>
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><a href="cayley.html#K2" title="CAYLEY:func.2">SymGroup</a> <span class="p2">(<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>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_2"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="matrix_2.html#D10" title="MATRIX_2:def.10">MATRIX_2:def 10</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>
<span class="kw">hence </span><a NAME="E4:16"/>
 <a href="cayley.html#K2" title="CAYLEY:func.2">SymGroup</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</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/>


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