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<b>thus </b><a NAME="E1:13_1_1"/>
( <font color="Maroon" title="c1">IT</font> is <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a> implies the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">IT</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c2">e</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">IT</font><b> such that </b><a NAME="E2:13_1_1_1"/><i><font color="Green" title="E7">A1</font></i>: 
for <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">IT</font> holds <br/> ( <font color="Olive" title="b1">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> &amp; <font color="Maroon" title="c2">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D2" title="GROUP_1:def.2">GROUP_1:def 2</a><br/>

<b>take </b>
<font color="Maroon" title="c2">e</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="setwiseo.html#D2" title="SETWISEO:def.2">SETWISEO:def 2</a><br/>

<a NAME="E3:13_1_1_1"/>
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">IT</font> holds <br/> ( the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">IT</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">e</font>,<font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">IT</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">IT</font>;<br/>

<b>thus </b>the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">IT</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c3">a</font> = 
<font color="Maroon" title="c2">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">a</font>

<br/>.= 
<font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="txt" href="monoid_0.html#E2:13_1_1_1"><i><font color="Green" title="E7">A1</font></i></a></i>
;<br/>

<b>thus </b>the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">IT</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">e</font> = 
<font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">e</font>

<br/>.= 
<font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="txt" href="monoid_0.html#E2:13_1_1_1"><i><font color="Green" title="E7">A1</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:13_1_1_1"/>
<font color="Maroon" title="c2">e</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">IT</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T11" title="BINOP_1:th.11">BINOP_1:11</a></i>;<br/>


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

<b>given </b><font color="Maroon" title="c2">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">IT</font><b> such that </b><a NAME="E3:13_1_1"/><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">IT</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="setwiseo.html#D2" title="SETWISEO:def.2">SETWISEO:def 2</a><br/>

<b>take </b>
<font color="Maroon" title="c2">x</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D2" title="GROUP_1:def.2">GROUP_1:def 2</a><br/>

<b>let </b><font color="Maroon" title="c3">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">IT</font>;<br/>

<b>thus </b><a NAME="E4:13_1_1"/>
( <font color="Maroon" title="c3">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> &amp; <font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E3:13_1_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="binop_1.html#T11" title="BINOP_1:th.11">BINOP_1:11</a></i>;<br/>


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