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

<span class="kw">let </span><font color="Maroon" title="c1">D</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">FlattenSeq</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <font color="Maroon" title="c1">D</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">g</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:110"/><span class="lab"><font color="Green" title="E65">A1</font></span>: 
 for <font color="Olive" title="b1">d1</font>, <font color="Olive" title="b2">d2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  holds  <font color="Maroon" title="c2">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>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d1</font> <a href="afinsq_1.html#K15" title="AFINSQ_1:func.15">^</a> <font color="Olive" title="b2">d2</font>
 <span class="kw">and </span><br/><a NAME="E3:110"/><span class="lab"><font color="Green" title="E66">A2</font></span>: 
 <a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">FlattenSeq</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#D10" target="_self" title="AFINSQ_2:def.10">Def11</a></span>;<br/>
<a NAME="E4:110"/><span class="lab"><font color="Green" title="E67">A3</font></span>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T43" title="AFINSQ_1:th.43">AFINSQ_1:43</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">p</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T43" title="AFINSQ_1:th.43">AFINSQ_1:43</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">g</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c4">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c2">g</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c4">a</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> )</span><br/><span class="kw">thus </span><font color="Maroon" title="c2">g</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c4">a</font>) = 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c4">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E4:110"><span class="lab"><font color="Green" title="E67">A3</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T29" title="AFINSQ_1:th.29">AFINSQ_1:29</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">g</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c4">a</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font></span><br/><span class="kw">thus </span><font color="Maroon" title="c2">g</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c4">a</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>) = 
<font color="Maroon" title="c4">a</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E4:110"><span class="lab"><font color="Green" title="E67">A3</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T29" title="AFINSQ_1:th.29">AFINSQ_1:29</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>
<a NAME="E7:110"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A4</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c2">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#T3" title="BINOP_1:th.3">BINOP_1:3</a></span>;<br/>
<a NAME="E8:110"/><span class="kw">then </span>
<font color="Maroon" title="c2">g</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setwiseo.html#D2" title="SETWISEO:def.2">SETWISEO:def 2</a></span>;<br/>
<a NAME="E9:110"/><span class="kw">then </span>
<font color="Maroon" title="c2">g</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a></span>)</span></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> <font color="Maroon" title="c2">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T72" target="_self" title="AFINSQ_2:th.72">Th84</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:110"/>
 <a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">FlattenSeq</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <font color="Maroon" title="c1">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E7:110"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="ref" href="binop_1.html#D8" title="BINOP_1:def.8">BINOP_1:def 8</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>
