<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">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> ;<br/><b>let </b><font color="Maroon" title="c2">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c3">F</font>, <font color="Maroon" title="c4">G</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">q</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:42"/><i><font color="Green" title="E20">A1</font></i>: 
( <font color="Maroon" title="c3">F</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Maroon" title="c3">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> )
 <b>and </b><br/><a NAME="E2:42"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a> 
 <b>and </b><br/><a NAME="E3:42"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c2">e</font> <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> <font color="Maroon" title="c3">F</font>
 <b>and </b><br/><a NAME="E4:42"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c4">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font>
 <b>and </b><br/><a NAME="E5:42"/><i><font color="Green" title="E24">A5</font></i>: 
for <font color="Olive" title="b1">d1</font>, <font color="Olive" title="b2">d2</font>, <font color="Olive" title="b3">d3</font>, <font color="Olive" title="b4">d4</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> holds  <font color="Maroon" title="c3">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">d2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b3">d3</font>,<font color="Olive" title="b4">d4</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">d1</font>,<font color="Olive" title="b3">d3</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b2">d2</font>,<font color="Olive" title="b4">d4</font></span>)</span>
 <b>and </b><br/><a NAME="E6:42"/><i><font color="Green" title="E25">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font> <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="c6">q</font>
 ;<br/>

<a NAME="E7:42"/><i><font color="Green" title="E26">A7</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="setwop_2.html#E6:42"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="finseq_2.html#T86" title="FINSEQ_2:th.86">FINSEQ_2:86</a></i>;<br/>
<a NAME="E8:42"/><i><font color="Green" title="E27">A8</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c5">p</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c6">q</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:42"/><i><font color="Green" title="E28">A9</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c5">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c6">q</font></span>)</span> = 
<font color="Maroon" title="c4">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p2">(<span class="default"><a href="finsop_1.html#K4" title="FINSOP_1:func.4">findom</a> <font color="Maroon" title="c5">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">e</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c6">q</font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:42"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="setwop_2.html#E2:42"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="setwop_2.html#E3:42"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="setwop_2.html#D2" target="_self" title="SETWOP_2:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c4">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p2">(<span class="default"><a href="finsop_1.html#K4" title="FINSOP_1:func.4">findom</a> <font color="Maroon" title="c5">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">e</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p2">(<span class="default"><a href="finsop_1.html#K4" title="FINSOP_1:func.4">findom</a> <font color="Maroon" title="c6">q</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">e</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:42"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="setwop_2.html#E2:42"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="setwop_2.html#E3:42"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="setwop_2.html#D2" target="_self" title="SETWOP_2:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="finsop_1.html#K4" title="FINSOP_1:func.4">findom</a> <font color="Maroon" title="c5">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">e</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">e</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:42"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="setwop_2.html#E2:42"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="setwop_2.html#E3:42"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="setwop_2.html#E4:42"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="setwop_2.html#E5:42"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="setwop_2.html#E6:42"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="setwop_2.html#E8:42"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="setwop_2.html#T11" target="_self" title="SETWOP_2:th.11">Th11</a></i>
<br/>.= 
<font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="finsop_1.html#K4" title="FINSOP_1:func.4">findom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font></span>)</span>,<font color="Maroon" title="c2">e</font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E4:42"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="setwop_2.html#E6:42"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="setwop_2.html#E7:42"><i><font color="Green" title="E26">A7</font></i></a>, <a class="txt" href="setwop_2.html#E8:42"><i><font color="Green" title="E27">A8</font></i></a>, <a class="txt" href="setwop_2.html#E9:42"><i><font color="Green" title="E28">A9</font></i></a>, <a class="txt" href="setwop_2.html#E31"><i><font color="Green" title="E12">Lm4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">F</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:42"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="setwop_2.html#E2:42"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="setwop_2.html#E3:42"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="setwop_2.html#D2" target="_self" title="SETWOP_2:def.2">Def2</a></i>
;<br/>


</div>
