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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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">D</font><br/> for <font color="Olive" title="b3">o</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Olive" title="b1">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="b3">o</font> holds <br/><font color="Maroon" title="c1">A</font> <a href="monoid_1.html#K5" title="MONOID_1:func.5">--&gt;</a> <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="b3">o</font>,<font color="Olive" title="b1">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">A</font></span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c2">D</font><br/> for <font color="Olive" title="b2">o</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">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="Olive" title="b2">o</font> holds <br/><font color="Maroon" title="c1">A</font> <a href="monoid_1.html#K5" title="MONOID_1:func.5">--&gt;</a> <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="Olive" title="b2">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><br/><b>let </b><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="c2">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c2">D</font>  st <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="Olive" title="b1">o</font> holds <br/><font color="Maroon" title="c1">A</font> <a href="monoid_1.html#K5" title="MONOID_1:func.5">--&gt;</a> <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="Olive" title="b1">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><br/><b>let </b><font color="Maroon" title="c4">o</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:()"><i><font color="Red">::  thesis: </font></i></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="c4">o</font> implies <font color="Maroon" title="c1">A</font> <a href="monoid_1.html#K5" title="MONOID_1:func.5">--&gt;</a> <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="c4">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><br/>







<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E9">A1</font></i>: 
<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="c4">o</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">A</font> <a href="monoid_1.html#K5" title="MONOID_1:func.5">--&gt;</a> <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="c4">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><br/>

<b>set </b><font color="Maroon" title="c5">e</font> = <font color="Maroon" title="c1">A</font> <a href="monoid_1.html#K5" title="MONOID_1:func.5">--&gt;</a> <font color="Maroon" title="c3">a</font>;<br/>
<b>set </b><font color="Maroon" title="c6">F</font> = <font color="Maroon" title="c4">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>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">f</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K4" title="BINOP_1:func.4">.</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> <font color="Maroon" title="c3">a</font></span>)</span>,<font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c7">f</font>,<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> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> )</span><br/><a NAME="E1:20_1"/><i><font color="Green" title="E10">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font> <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/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K4" title="BINOP_1:func.4">.</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> <font color="Maroon" title="c3">a</font></span>)</span>,<font color="Maroon" title="c7">f</font> = 
<font color="Maroon" title="c4">o</font> <a href="monoid_1.html#K3" title="MONOID_1:func.3">.:</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> <font color="Maroon" title="c3">a</font></span>)</span>,<font color="Maroon" title="c7">f</font>
<b>by </b><i><a class="ref" href="monoid_1.html#D2" target="_self" title="MONOID_1:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c4">o</font> <a href="monoid_1.html#K6" title="MONOID_1:func.6">[;]</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">f</font>
<b>by </b><i><a class="txt" href="monoid_1.html#E1:20_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="funcop_1.html#T41" title="FUNCOP_1:th.41">FUNCOP_1:41</a></i>
<br/>.= 
<font color="Maroon" title="c7">f</font>
<b>by </b><i><a class="txt" href="monoid_1.html#E1:20"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="monoid_1.html#T6" target="_self" title="MONOID_1:th.6">Th6</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c7">f</font>,<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> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font></span><br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c7">f</font>,<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> <font color="Maroon" title="c3">a</font></span>)</span> = 
<font color="Maroon" title="c4">o</font> <a href="monoid_1.html#K3" title="MONOID_1:func.3">.:</a> <font color="Maroon" title="c7">f</font>,<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> <font color="Maroon" title="c3">a</font></span>)</span>
<b>by </b><i><a class="ref" href="monoid_1.html#D2" target="_self" title="MONOID_1:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c4">o</font> <a href="monoid_1.html#K7" title="MONOID_1:func.7">[:]</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="txt" href="monoid_1.html#E1:20_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="funcop_1.html#T34" title="FUNCOP_1:th.34">FUNCOP_1:34</a></i>
<br/>.= 
<font color="Maroon" title="c7">f</font>
<b>by </b><i><a class="txt" href="monoid_1.html#E1:20"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="monoid_1.html#T6" target="_self" title="MONOID_1:th.6">Th6</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:20"/>
<font color="Maroon" title="c1">A</font> <a href="monoid_1.html#K5" title="MONOID_1:func.5">--&gt;</a> <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="c4">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>
 <b>by </b><i><a class="ref" href="binop_1.html#T11" title="BINOP_1:th.11">BINOP_1:11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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