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<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">B</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="binop_1.html#NM1">UnOp</a> of <font color="Maroon">D</font>;<br/>







<a NAME="E1:19"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:19"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">B</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  &amp; <font color="Maroon">B</font> is <a href="binop_1.html#V2">associative</a> &amp; <font color="Maroon">C</font> <a href="finseqop.html#R1">is_an_inverseOp_wrt</a> <font color="Maroon">B</font> )
 ;<br/>

<a NAME="E3:19"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">B</font> <a href="finseqop.html#NR2" target="_self">has_an_inverseOp</a> 
 
<b>by </b><i><a class="ref" href="finseqop.html#D2">FINSEQOP:def 2</a></i>;<br/>
<a NAME="E4:19"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">C</font> <a href="funct_2.html#R2">=</a>  <a href="finseqop.html#K6">the_inverseOp_wrt</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="finseqop.html#D3">FINSEQOP:def 3</a></i>;<br/>
<a NAME="E5:19"/>
 ex <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> st <font color="Olive">d</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="setwiseo.html#D2">SETWISEO:def 2</a></i>;<br/>
<a NAME="E6:19"/><b>then </b>
 <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">B</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="binop_1.html#D8">BINOP_1:def 8</a></i>;<br/>
<a NAME="E7:19"/><b>then </b>
<font color="Maroon">n</font> <a href="prvect_1.html#K4" target="_self">.--&gt;</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">B</font></span>)</span> <a href="binop_1.html#R3">is_a_unity_wrt</a>  <a href="prvect_1.html#K2" target="_self">product</a> <font color="Maroon">B</font>,<font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:19"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K2" target="_self">product</a> <font color="Maroon">B</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="binop_1.html#D8">BINOP_1:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>;<br/><b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> ;<br/><b>reconsider </b><font color="Maroon">cf</font> = <font color="Maroon">C</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">f'</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="finseq_2.html#T133">FINSEQ_2:133</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="prvect_1.html#K2" target="_self">product</a> <font color="Maroon">B</font>,<font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K3" target="_self">product</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="prvect_1.html#K2" target="_self">product</a> <font color="Maroon">B</font>,<font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">f'</font>,<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">f'</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">B</font> <a href="prvect_1.html#K1" target="_self">.:</a> <font color="Maroon">f'</font>,<font color="Maroon">cf</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="finseqop.html#T77">FINSEQOP:77</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="prvect_1.html#K2" target="_self">product</a> <font color="Maroon">B</font>,<font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K3" target="_self">product</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="prvect_1.html#K2" target="_self">product</a> <font color="Maroon">B</font>,<font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">f'</font></span>)</span>,<font color="Maroon">f'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">B</font> <a href="prvect_1.html#K1" target="_self">.:</a> <font color="Maroon">cf</font>,<font color="Maroon">f'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="finseqop.html#T77">FINSEQOP:77</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:19"/>
 <a href="prvect_1.html#K3" target="_self">product</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> <a href="finseqop.html#R1">is_an_inverseOp_wrt</a>  <a href="prvect_1.html#K2" target="_self">product</a> <font color="Maroon">B</font>,<font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="prvect_1.html#T3" target="_self">Th3</a>, <a class="ref" href="finseqop.html#D1">FINSEQOP:def 1</a></i>;<br/>


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