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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">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">o</font> is <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a> holds <br/>(  <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="b2">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>)</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="Olive" title="b2">o</font></span>)</span> &amp; <font color="Olive" title="b2">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> is <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a> )</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">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">o</font> is <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a> holds <br/>(  <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>,<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="Olive" title="b1">o</font></span>)</span> &amp; <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> is <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">o</font> is <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a> implies (  <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> &amp; <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> is <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a> ) )</span><br/>





<b>given </b><font color="Maroon" title="c4">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D</font><b> such that </b><a NAME="E2:21"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c3">o</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="setwiseo.html#D2" title="SETWISEO:def.2">SETWISEO:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="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> &amp; <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> is <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a> )</span><br/>

<a NAME="E3:21"/>
( <font color="Maroon" title="c4">a</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> &amp; <font color="Maroon" title="c1">A</font> <a href="monoid_1.html#K5" title="MONOID_1:func.5">--&gt;</a> <font color="Maroon" title="c4">a</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <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> )
 
<b>by </b><i><a class="txt" href="#E2:21"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="monoid_1.html#T10" target="_self" title="MONOID_1:th.10">Th10</a>, <a class="ref" href="binop_1.html#D8" title="BINOP_1:def.8">BINOP_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E4:21"/>
 <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="ref" href="binop_1.html#D8" title="BINOP_1:def.8">BINOP_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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> is <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a></span><br/>

<b>take </b>
<font color="Maroon" title="c1">A</font> <a href="monoid_1.html#K5" title="MONOID_1:func.5">--&gt;</a> <font color="Maroon" title="c4">a</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="setwiseo.html#D2" title="SETWISEO:def.2">SETWISEO:def 2</a> <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="c4">a</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <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><br/>

<b>thus </b><a NAME="E5:21"/>
<font color="Maroon" title="c1">A</font> <a href="monoid_1.html#K5" title="MONOID_1:func.5">--&gt;</a> <font color="Maroon" title="c4">a</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <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>
 <b>by </b><i><a class="txt" href="#E2:21"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="monoid_1.html#T10" target="_self" title="MONOID_1:th.10">Th10</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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