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<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/>



<b>thus </b><a NAME="E1:10"/>
( <font color="Maroon">f</font> is <a href="monoid_0.html#V5" target="_self">invertible</a> implies ( <font color="Maroon">f</font> is <a href="monoid_0.html#V3" target="_self">left-invertible</a> &amp; <font color="Maroon">f</font> is <a href="monoid_0.html#V4" target="_self">right-invertible</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  ex <font color="Olive">r</font>, <font color="Olive">l</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> st <br/>( <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">l</font>,<font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D5" target="_self">MONOID_0:def 5</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>, <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>, <font color="Maroon">l</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E2:10_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">l</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">l</font> = <font color="Maroon">l</font>;<br/><b>thus </b><a NAME="E3:10_1_1"/>
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">l</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  ex <font color="Olive">l</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> st <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">l</font>,<font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D3" target="_self">MONOID_0:def 3</a><br/>

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>, <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D4" target="_self">MONOID_0:def 4</a><br/>

<b>consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>, <font color="Maroon">l</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E5:10_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">l</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">r</font>
;<br/>

<b>thus </b><a NAME="E6:10_1"/>
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E2:10"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  ex <font color="Olive">l</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> st <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">l</font>,<font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font>
 <b>and </b><br/><a NAME="E3:10"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  ex <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> st <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D3" target="_self">MONOID_0:def 3</a>,<a class="ref" href="monoid_0.html#D4" target="_self">MONOID_0:def 4</a><br/>

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>, <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D5" target="_self">MONOID_0:def 5</a><br/>

<b>consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E5:10"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">l</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E7:10"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">r</font>
;<br/>

<b>take </b>
<font color="Maroon">l</font>
;<br/>

<b>thus </b><a NAME="E8:10"/>
( <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">l</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, </i>;<br/>


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