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<b>let </b><font color="Maroon">M</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L1">multLoopStr</a> ;<br/>

<a NAME="E1:37"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">M</font> is <a href="vectsp_2.html#V1">well-unital</a> iff <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">M</font>) <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">M</font>) )
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E2:37"/>
( <font color="Maroon">M</font> is <a href="vectsp_2.html#V1">well-unital</a> implies for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> holds <br/> ( <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">M</font>) <a href="group_1.html#K1">*</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">M</font>) <a href="hidden.html#R1">=</a> <font color="Olive">a</font> ) )
 <b>by </b><i>, <a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>;<br/>

<b>assume </b><a NAME="E3:37"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> holds <br/> ( <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">M</font>) <a href="group_1.html#K1">*</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">M</font>) <a href="hidden.html#R1">=</a> <font color="Olive">a</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </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">M</font>;<br/><a NAME="E1:37_1"/>
( <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">M</font>) <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">M</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="monoid_0.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E2:37_1"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">M</font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">M</font>),<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">M</font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">M</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:37"/>
<font color="Maroon">M</font> is <a href="vectsp_2.html#V1">well-unital</a>
 <b>by </b><i>, <a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>;<br/>


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