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<b>let </b><font color="Maroon">G1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V1">strict</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="monoid_0.html#V2" target="_self">constituted-FinSeqs</a> <a href="monoid_0.html#V16" target="_self">cancelable</a> <a href="monoid_0.html#V17" target="_self">uniquely-decomposable</a>  <a href="group_1.html#L1">HGrStr</a> , <font color="Maroon">G2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V1">strict</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="monoid_0.html#V2" target="_self">constituted-FinSeqs</a> <a href="monoid_0.html#V16" target="_self">cancelable</a> <a href="monoid_0.html#V17" target="_self">uniquely-decomposable</a>  <a href="group_1.html#L1">HGrStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:126_1_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G1</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a>  &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font> holds  <font color="Olive">p</font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">q</font> ) )
 <b>and </b><br/><a NAME="E2:126_1_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G2</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a>  &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font> holds  <font color="Olive">p</font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">q</font> ) )
 ;<br/>

<b>set </b><font color="Maroon">f1</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G1</font>);<br/>
<b>set </b><font color="Maroon">f2</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G2</font>);<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="126_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a> , <font color="Maroon">b</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a> ;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">a</font>, <font color="Maroon">q</font> = <font color="Maroon">b</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a>  ;<br/><b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font> <b>by </b><i><a class="txt" href="monoid_0.html#E28"><i><font color="Green">Lemma58</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">a2</font> = <font color="Maroon">a</font>, <font color="Maroon">b2</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font> <b>by </b><i><a class="ref" href="monoid_0.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E4:126_1_2_1"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G2</font>) <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="monoid_0.html#K1" target="_self">^</a> <font color="Maroon">q</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G1</font>) <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a1</font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="monoid_0.html#K1" target="_self">^</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E28"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="monoid_0.html#T9" target="_self">Th9</a></i>;<br/><b>hence </b><a NAME="E5:126_1_2_1"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G1</font>) <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G2</font>) <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:126_1_2"/>
<font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font>
 <b>by </b><i><a class="txt" href="monoid_0.html#E28"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="monoid_0.html#T9" target="_self">Th9</a>, <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


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