<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">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="c2">f</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>;<br/>



<b>thus </b><a NAME="E1:10"/>
( <font color="Maroon" title="c2">f</font> is <a href="monoid_0.html#V5" title="MONOID_0:attr.5">invertible</a> implies ( <font color="Maroon" title="c2">f</font> is <a href="monoid_0.html#V3" title="MONOID_0:attr.3">left-invertible</a> &amp; <font color="Maroon" title="c2">f</font> is <a href="monoid_0.html#V4" title="MONOID_0:attr.4">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" title="E5">A1</font></i>: 
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="c1">D</font>  ex <font color="Olive" title="b3">r</font>, <font color="Olive" title="b4">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> st <br/>( <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b4">l</font>,<font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D5" target="_self" title="MONOID_0:def.5">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" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>;<br/><b>consider </b><font color="Maroon" title="c5">r</font>, <font color="Maroon" title="c6">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font><b> such that </b><br/><a NAME="E2:10_1_1"/><i><font color="Green" title="E6">A2</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E1:10_1"><i><font color="Green" title="E5">A1</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c7">l</font> = <font color="Maroon" title="c6">l</font>;<br/><b>thus </b><a NAME="E3:10_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">l</font>,<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="txt" href="monoid_0.html#E2:10_1_1"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_1"/>
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="c1">D</font>  ex <font color="Olive" title="b3">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> st <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b3">l</font>,<font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D3" target="_self" title="MONOID_0:def.3">MONOID_0:def 3</a><br/>

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D4" target="_self" title="MONOID_0:def.4">MONOID_0:def 4</a><br/>

<b>consider </b><font color="Maroon" title="c5">r</font>, <font color="Maroon" title="c6">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font><b> such that </b><br/><a NAME="E5:10_1"/><i><font color="Green" title="E6">A3</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E1:10_1"><i><font color="Green" title="E5">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">r</font>
;<br/>

<b>thus </b><a NAME="E6:10_1"/>
<font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="txt" href="monoid_0.html#E5:10_1"><i><font color="Green" title="E6">A3</font></i></a></i>;<br/>


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

<b>assume </b><b>that </b><br/><a NAME="E2:10"/><i><font color="Green" title="E5">A4</font></i>: 
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="c1">D</font>  ex <font color="Olive" title="b3">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> st <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b3">l</font>,<font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font>
 <b>and </b><br/><a NAME="E3:10"/><i><font color="Green" title="E6">A5</font></i>: 
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="c1">D</font>  ex <font color="Olive" title="b3">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> st <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D3" target="_self" title="MONOID_0:def.3">MONOID_0:def 3</a>,<a class="ref" href="monoid_0.html#D4" target="_self" title="MONOID_0:def.4">MONOID_0:def 4</a><br/>

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D5" target="_self" title="MONOID_0:def.5">MONOID_0:def 5</a><br/>

<b>consider </b><font color="Maroon" title="c5">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font><b> such that </b><br/><a NAME="E5:10"/><i><font color="Green" title="E7">A6</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">l</font>,<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="txt" href="monoid_0.html#E2:10"><i><font color="Green" title="E5">A4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">r</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font><b> such that </b><br/><a NAME="E7:10"/><i><font color="Green" title="E8">A7</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="txt" href="monoid_0.html#E3:10"><i><font color="Green" title="E6">A5</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">r</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c5">l</font>
;<br/>

<b>thus </b><a NAME="E8:10"/>
( <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">l</font>,<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E5:10"><i><font color="Green" title="E7">A6</font></i></a>, <a class="txt" href="monoid_0.html#E7:10"><i><font color="Green" title="E8">A7</font></i></a></i>;<br/>


</div>
