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

<span class="kw">let </span><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> ; <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">o</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b2">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b2">a</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Olive" title="b1">o</font> holds <br/>( <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Olive" title="b1">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a>  &amp; <font color="Olive" title="b1">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a>  is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp;  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">o</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">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">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b1">a</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c2">o</font> holds <br/>( <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a>  &amp; <font color="Maroon" title="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a>  is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp;  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">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 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">a</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c2">o</font> implies ( <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a>  &amp; <font color="Maroon" title="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a>  is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp;  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:81"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c2">o</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"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a>  &amp; <font color="Maroon" title="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a>  is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp;  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">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"><font color="Maroon" title="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a> </span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a> </span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> )</span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a> </span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">x</font> = 
<font color="Maroon" title="c2">o</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">x</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_1.html#T45" target="_self" title="MONOID_1:th.45">Th45</a></span>
<br/>.= 
<font color="Maroon" title="c1">D</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="monoid_1.html#T52" target="_self" title="MONOID_1:th.52">Th52</a></span>
<br/>.= 
<font color="Maroon" title="c4">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</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"><font color="Maroon" title="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a> </span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a> </span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> = 
<font color="Maroon" title="c2">o</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">x</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_1.html#T45" target="_self" title="MONOID_1:th.45">Th45</a></span>
<br/>.= 
<font color="Maroon" title="c1">D</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="monoid_1.html#T52" target="_self" title="MONOID_1:th.52">Th52</a></span>
<br/>.= 
<font color="Maroon" title="c4">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</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>
<span class="kw">hence </span><a NAME="E3:81"/><span class="lab"><font color="Green" title="E49">A2</font></span>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#T11" title="BINOP_1:th.11">BINOP_1:11</a></span>; <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="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a>  is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp;  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/>

<span class="kw">hence </span><a NAME="E4:81"/>
 ex <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font> st <font color="Olive" title="b1">A</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="setwiseo.html#D2" title="SETWISEO:def.2">SETWISEO:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<span class="kw">thus </span><a NAME="E5:81"/>
 <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:81"><span class="lab"><font color="Green" title="E49">A2</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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
