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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="vectsp_2.html#V1">well-unital</a>  <a href="vectsp_1.html#L1">multLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="monoid_0.html#M3" target="_self">MonoidalSubStr</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:82"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is    <a href="monoid_0.html#M2" target="_self">SubStr</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="monoid_0.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E2:82"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>) &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) )
 
<b>by </b><i><a class="ref" href="monoid_0.html#D21" target="_self">Def21</a>, <a class="ref" href="monoid_0.html#D24" target="_self">Def24</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:82_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>)
 <b>by </b><i><a class="ref" href="monoid_0.html#T25" target="_self">Th25</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>thus </b><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>2</sub></font>),<font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font>

<br/>.= 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="monoid_0.html#E1:82"><i><font color="Green">E42</font></i></a>, <a class="txt" href="monoid_0.html#E2:82"><i><font color="Green">E43</font></i></a>, <a class="ref" href="monoid_0.html#T27" target="_self">Th27</a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>),<font color="Maroon">c<sub>4</sub></font>

<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="monoid_0.html#E2:82"><i><font color="Green">E43</font></i></a>, <a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>
;<br/><b>thus </b><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>2</sub></font>) = 
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>2</sub></font>)

<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>)
<b>by </b><i><a class="txt" href="monoid_0.html#E1:82"><i><font color="Green">E42</font></i></a>, <a class="txt" href="monoid_0.html#E2:82"><i><font color="Green">E43</font></i></a>, <a class="ref" href="monoid_0.html#T27" target="_self">Th27</a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>)

<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="monoid_0.html#E2:82"><i><font color="Green">E43</font></i></a>, <a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:82"/>
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>)
 <b>by </b><i><a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D21" target="_self">MONOID_0:def 21</a><br/>


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