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

<b>thus </b><a NAME="E1:13_1_1"/>
( <font color="Maroon">IT</font> is <a href="group_1.html#V2">unital</a> implies the <a href="group_1.html#U1">mult</a> of <font color="Maroon">IT</font> <a href="setwiseo.html#NR1" target="_self">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">e</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">IT</font><b> such that </b><a NAME="E1:13_1_1_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">h</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">IT</font> holds <br/> ( <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp; <font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D2">GROUP_1:def 2</a><br/>

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

<a NAME="E2:13_1_1_1"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">IT</font> holds <br/> ( the <a href="group_1.html#U1">mult</a> of <font color="Maroon">IT</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">e</font>,<font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; the <a href="group_1.html#U1">mult</a> of <font color="Maroon">IT</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">IT</font>;<br/>

<b>thus </b>the <a href="group_1.html#U1">mult</a> of <font color="Maroon">IT</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">e</font>,<font color="Maroon">a</font> = 
<font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>

<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i/>
;<br/>

<b>thus </b>the <a href="group_1.html#U1">mult</a> of <font color="Maroon">IT</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">e</font> = 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font>

<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:13_1_1_1"/>
<font color="Maroon">e</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">IT</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>;<br/>


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

<b>given </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">IT</font><b> such that </b><a NAME="E2:13_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">IT</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="setwiseo.html#D2">SETWISEO:def 2</a><br/>

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

<b>let </b><font color="Maroon">h</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">IT</font>;<br/>

<a NAME="E3:13_1_1"/>
for <font color="Olive">G</font> being non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a> <br/> for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">G</font> holds <br/> ( the <a href="group_1.html#U1">mult</a> of <font color="Maroon">IT</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> &amp; the <a href="group_1.html#U1">mult</a> of <font color="Maroon">IT</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">h</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> )
 
<b>by </b><i>, <a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>;<br/>
<b>hence </b><a NAME="E4:13_1_1"/>
( <font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> &amp; <font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> )
 ;<br/>


</div>
