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<b>let </b><font color="Maroon" title="c1">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><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> ;<br/><b>let </b><font color="Maroon" title="c3">o</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D</font>;<br/>





<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c3">o</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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="Maroon" title="c3">o</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</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">o</font> holds <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b2">b</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">o</font> ) ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D9" title="MONOID_0:def.9">MONOID_0:def 9</a><br/>

<b>hence </b><a NAME="E2:25"/>
<font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">A</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a> 
 <b>by </b><i><a class="ref" href="monoid_1.html#T11" target="_self" title="MONOID_1:th.11">Th11</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D9" title="MONOID_0:def.9">MONOID_0:def 9</a><br/>

<b>let </b><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">D</font>;<br/>

<b>assume </b><a NAME="E3:25"/><i><font color="Green" title="E15">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">A</font></span>)</span>
 ;<br/>

<a NAME="E4:25"/><i><font color="Green" title="E16">A3</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o</font> <a href="monoid_1.html#K3" title="MONOID_1:func.3">.:</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font> &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="c3">o</font>,<font color="Maroon" title="c2">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> <a href="monoid_1.html#K5" title="MONOID_1: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">o</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="monoid_1.html#E1:25"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="monoid_1.html#D2" target="_self" title="MONOID_1:def.2">Def2</a>, <a class="ref" href="monoid_1.html#T11" target="_self" title="MONOID_1:th.11">Th11</a></i>;<br/>
<a NAME="E5:25"/><i><font color="Green" title="E17">A4</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="monoid_1.html#K5" title="MONOID_1:func.5">--&gt;</a> <span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c3">o</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font> <a href="monoid_1.html#K3" title="MONOID_1:func.3">.:</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:25_1"/><i><font color="Green" title="E18">A5</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D</font> <b>by </b><i><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/><a NAME="E3:25_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font> <a href="monoid_1.html#K3" title="MONOID_1:func.3">.:</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="monoid_1.html#K5" title="MONOID_1:func.5">--&gt;</a> <span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c3">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</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">o</font> )
 <b>by </b><i><a class="txt" href="monoid_1.html#E5:25"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="monoid_1.html#E1:25_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a>, <a class="ref" href="funcop_1.html#T28" title="FUNCOP_1:th.28">FUNCOP_1:28</a></i>;<br/><b>hence </b><a NAME="E4:25_1"/>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="monoid_1.html#E1:25"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="monoid_1.html#E3:25"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="monoid_1.html#E4:25"><i><font color="Green" title="E16">A3</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:25"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">g</font>
 <b>by </b><i><a class="txt" href="monoid_1.html#E5:25"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:25_2"/><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D</font> <b>by </b><i><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/><a NAME="E3:25_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font> <a href="monoid_1.html#K3" title="MONOID_1:func.3">.:</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="monoid_1.html#K5" title="MONOID_1:func.5">--&gt;</a> <span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c3">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</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">o</font> )
 <b>by </b><i><a class="txt" href="monoid_1.html#E5:25"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="monoid_1.html#E1:25_2"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a>, <a class="ref" href="funcop_1.html#T28" title="FUNCOP_1:th.28">FUNCOP_1:28</a></i>;<br/><b>hence </b><a NAME="E4:25_2"/>
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="monoid_1.html#K5" title="MONOID_1:func.5">--&gt;</a> <span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c3">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="monoid_1.html#E1:25"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="monoid_1.html#E3:25"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="monoid_1.html#E4:25"><i><font color="Green" title="E16">A3</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:25"/>
<font color="Maroon" title="c5">g</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="monoid_1.html#E4:25"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="monoid_1.html#E5:25"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


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