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<b>let </b><font color="Maroon" title="c2">G1</font>, <font color="Maroon" title="c3">G2</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> <a href="monoid_0.html#V1" title="MONOID_0:attr.1">constituted-Functions</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:136_1_2"/><i><font color="Green" title="E62">A6</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">G1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G1</font> holds  <font color="Olive" title="b1">f</font> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Olive" title="b2">g</font> ) )
 <b>and </b><br/><a NAME="E2:136_1_2"/><i><font color="Green" title="E63">A7</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">G2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G2</font> holds  <font color="Olive" title="b1">f</font> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Olive" title="b2">g</font> ) )
 ;<br/>

<b>set </b><font color="Maroon" title="c4">f1</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">G1</font>);<br/>
<b>set </b><font color="Maroon" title="c5">f2</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">G2</font>);<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G1</font>;<br/><b>reconsider </b><font color="Maroon" title="c8">a'</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c9">b'</font> = <font color="Maroon" title="c7">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G2</font> <b>by </b><i><a class="txt" href="monoid_0.html#E1:136_1_2"><i><font color="Green" title="E62">A6</font></i></a>, <a class="txt" href="monoid_0.html#E2:136_1_2"><i><font color="Green" title="E63">A7</font></i></a></i>;<br/><a NAME="E2:136_1_2_1"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">G1</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c6">a</font> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">G2</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a'</font> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <font color="Maroon" title="c9">b'</font> &amp; <font color="Maroon" title="c8">a'</font> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <font color="Maroon" title="c9">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a'</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c9">b'</font> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E1:136_1_2"><i><font color="Green" title="E62">A6</font></i></a>, <a class="txt" href="monoid_0.html#E2:136_1_2"><i><font color="Green" title="E63">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E3:136_1_2_1"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">G1</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">G2</font>) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:136_1_2"/>
<font color="Maroon" title="c2">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G2</font>
 <b>by </b><i><a class="txt" href="monoid_0.html#E1:136_1_2"><i><font color="Green" title="E62">A6</font></i></a>, <a class="txt" href="monoid_0.html#E2:136_1_2"><i><font color="Green" title="E63">A7</font></i></a>, <a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></i>;<br/>


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