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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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">D</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">B</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Olive" title="b1">D</font><br/> for <font color="Olive" title="b3">C</font> being  <a href="binop_1.html#NM1" title="BINOP_1:NM.1">UnOp</a> of <font color="Olive" title="b1">D</font>  st <font color="Olive" title="b2">B</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Olive" title="b2">B</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; <font color="Olive" title="b3">C</font> <a href="finseqop.html#R1" title="FINSEQOP:pred.1">is_an_inverseOp_wrt</a> <font color="Olive" title="b2">B</font> holds <br/> <a href="prvect_1.html#K3" title="PRVECT_1:func.3">product</a> <font color="Olive" title="b3">C</font>,<font color="Maroon" title="c1">n</font> <a href="finseqop.html#R1" title="FINSEQOP:pred.1">is_an_inverseOp_wrt</a>  <a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Olive" title="b2">B</font>,<font color="Maroon" title="c1">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">D</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">B</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D</font><br/> for <font color="Olive" title="b2">C</font> being  <a href="binop_1.html#NM1" title="BINOP_1:NM.1">UnOp</a> of <font color="Maroon" title="c2">D</font>  st <font color="Olive" title="b1">B</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Olive" title="b1">B</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; <font color="Olive" title="b2">C</font> <a href="finseqop.html#R1" title="FINSEQOP:pred.1">is_an_inverseOp_wrt</a> <font color="Olive" title="b1">B</font> holds <br/> <a href="prvect_1.html#K3" title="PRVECT_1:func.3">product</a> <font color="Olive" title="b2">C</font>,<font color="Maroon" title="c1">n</font> <a href="finseqop.html#R1" title="FINSEQOP:pred.1">is_an_inverseOp_wrt</a>  <a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Olive" title="b1">B</font>,<font color="Maroon" title="c1">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">B</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</font> being  <a href="binop_1.html#NM1" title="BINOP_1:NM.1">UnOp</a> of <font color="Maroon" title="c2">D</font>  st <font color="Maroon" title="c3">B</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Maroon" title="c3">B</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; <font color="Olive" title="b1">C</font> <a href="finseqop.html#R1" title="FINSEQOP:pred.1">is_an_inverseOp_wrt</a> <font color="Maroon" title="c3">B</font> holds <br/> <a href="prvect_1.html#K3" title="PRVECT_1:func.3">product</a> <font color="Olive" title="b1">C</font>,<font color="Maroon" title="c1">n</font> <a href="finseqop.html#R1" title="FINSEQOP:pred.1">is_an_inverseOp_wrt</a>  <a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">C</font> be   <a href="binop_1.html#NM1" title="BINOP_1:NM.1">UnOp</a> of <font color="Maroon" title="c2">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">B</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Maroon" title="c3">B</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; <font color="Maroon" title="c4">C</font> <a href="finseqop.html#R1" title="FINSEQOP:pred.1">is_an_inverseOp_wrt</a> <font color="Maroon" title="c3">B</font> implies  <a href="prvect_1.html#K3" title="PRVECT_1:func.3">product</a> <font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c1">n</font> <a href="finseqop.html#R1" title="FINSEQOP:pred.1">is_an_inverseOp_wrt</a>  <a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">n</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:27"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
<font color="Maroon" title="c3">B</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a> 
 <span class="kw">and </span><br/><a NAME="E2:27"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
<font color="Maroon" title="c3">B</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> 
 <span class="kw">and </span><br/><a NAME="E3:27"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
<font color="Maroon" title="c4">C</font> <a href="finseqop.html#R1" title="FINSEQOP:pred.1">is_an_inverseOp_wrt</a> <font color="Maroon" title="c3">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="prvect_1.html#K3" title="PRVECT_1:func.3">product</a> <font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c1">n</font> <a href="finseqop.html#R1" title="FINSEQOP:pred.1">is_an_inverseOp_wrt</a>  <a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">n</font></span><br/>

<a NAME="E4:27"/><span class="lab"><font color="Green" title="E18">A4</font></span>: 
<font color="Maroon" title="c3">B</font> is  <a href="finseqop.html#V1" title="FINSEQOP:attr.1">having_an_inverseOp</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="finseqop.html#D2" title="FINSEQOP:def.2">FINSEQOP:def 2</a></span>;<br/>
<a NAME="E5:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A5</font></span>: 
<font color="Maroon" title="c4">C</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="finseqop.html#K6" title="FINSEQOP:func.6">the_inverseOp_wrt</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="finseqop.html#D3" title="FINSEQOP:def.3">FINSEQOP:def 3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E20">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">f</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c2">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<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">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K3" title="PRVECT_1:func.3">product</a> <font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c3">B</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K3" title="PRVECT_1:func.3">product</a> <font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c3">B</font></span>)</span> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c6">f9</font> = <font color="Maroon" title="c5">f</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c2">D</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">cf</font> = <font color="Maroon" title="c4">C</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">f9</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c2">D</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T133" title="FINSEQ_2:th.133">FINSEQ_2:133</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<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">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K3" title="PRVECT_1:func.3">product</a> <font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<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="c6">f9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">f9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_1.html#D2" target="_self" title="PRVECT_1:def.2">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c3">B</font> <a href="prvect_1.html#K1" title="PRVECT_1:func.1">.:</a> <font color="Maroon" title="c6">f9</font>,<font color="Maroon" title="c7">cf</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_1.html#D1" target="_self" title="PRVECT_1:def.1">Def1</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c3">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E4:27"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="ref" href="finseqop.html#T77" title="FINSEQOP:th.77">FINSEQOP:77</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K3" title="PRVECT_1:func.3">product</a> <font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c3">B</font></span>)</span></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K3" title="PRVECT_1:func.3">product</a> <font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c5">f</font> = 
<span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">f9</font></span>)</span>,<font color="Maroon" title="c6">f9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_1.html#D2" target="_self" title="PRVECT_1:def.2">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c3">B</font> <a href="prvect_1.html#K1" title="PRVECT_1:func.1">.:</a> <font color="Maroon" title="c7">cf</font>,<font color="Maroon" title="c6">f9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_1.html#D1" target="_self" title="PRVECT_1:def.1">Def1</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c3">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E4:27"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="ref" href="finseqop.html#T77" title="FINSEQOP:th.77">FINSEQOP:77</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:27"/>
 ex <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D</font> st <font color="Olive" title="b1">d</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="setwiseo.html#D2" title="SETWISEO:def.2">SETWISEO:def 2</a></span>;<br/>
<a NAME="E8:27"/><span class="kw">then </span>
 <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c3">B</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D8" title="BINOP_1:def.8">BINOP_1:def 8</a></span>;<br/>
<a NAME="E9:27"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">.--&gt;</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a>  <a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_1.html#T16" target="_self" title="PRVECT_1:th.16">Th16</a></span>;<br/>
<a NAME="E10:27"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D8" title="BINOP_1:def.8">BINOP_1:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:27"/>
 <a href="prvect_1.html#K3" title="PRVECT_1:func.3">product</a> <font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c1">n</font> <a href="finseqop.html#R1" title="FINSEQOP:pred.1">is_an_inverseOp_wrt</a>  <a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:27"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="finseqop.html#D1" title="FINSEQOP:def.1">FINSEQOP:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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