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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><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="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/><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="c2">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">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/><span class="kw">let </span><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:()"><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="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/>







<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">assume </span><a NAME="E1:21"/><span class="lab"><font color="Green" title="E9">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="c4">o</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="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/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">f</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">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="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="E1:21_1"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><span class="kw">thus </span><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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_1.html#D2" target="_self" title="MONOID_1:def.2">Def2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="funcop_1.html#T41" title="FUNCOP_1:th.41">FUNCOP_1:41</a></span>
<br/>.= 
<font color="Maroon" title="c7">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="monoid_1.html#T6" target="_self" title="MONOID_1:th.6">Th6</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="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/><span class="kw">thus </span><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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_1.html#D2" target="_self" title="MONOID_1:def.2">Def2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="funcop_1.html#T34" title="FUNCOP_1:th.34">FUNCOP_1:34</a></span>
<br/>.= 
<font color="Maroon" title="c7">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="monoid_1.html#T6" target="_self" title="MONOID_1:th.6">Th6</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="E3:21"/>
<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 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"> verum</span><br/>


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