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<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">G</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c4">f2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c1">D</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c5">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>;<br/>







<b>reconsider </b><font color="Maroon" title="c6">g1</font> = <font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c7">g2</font> = <font color="Maroon" title="c4">f2</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c2">G</font>) <b>by </b><i><a class="ref" href="monoid_1.html#T18" target="_self" title="MONOID_1:th.18">Th18</a></i>;<br/>
<a NAME="E2:38"/>
( <font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c1">D</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">G</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c2">G</font>) <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">D</font> &amp; <font color="Maroon" title="c3">f1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c1">D</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font> )
 
<b>by </b><i><a class="ref" href="monoid_1.html#T18" target="_self" title="MONOID_1:th.18">Th18</a>, <a class="ref" href="group_1.html#D1" title="GROUP_1:def.1">GROUP_1:def 1</a></i>;<br/>
<a NAME="E3:38"/><b>then </b>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">G</font>) <a href="monoid_1.html#K3" title="MONOID_1:func.3">.:</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">D</font> &amp; <font color="Maroon" title="c3">f1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">G</font>) <a href="monoid_1.html#K3" title="MONOID_1:func.3">.:</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c7">g2</font> )
 
<b>by </b><i><a class="ref" href="monoid_1.html#D2" target="_self" title="MONOID_1:def.2">Def2</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="monoid_1.html#K10" title="MONOID_1:func.10">.</a> <font color="Maroon" title="c5">a</font> = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">G</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="monoid_1.html#K10" title="MONOID_1:func.10">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="monoid_1.html#K10" title="MONOID_1:func.10">.</a> <font color="Maroon" title="c5">a</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T28" title="FUNCOP_1:th.28">FUNCOP_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="monoid_1.html#K10" title="MONOID_1:func.10">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="monoid_1.html#K10" title="MONOID_1:func.10">.</a> <font color="Maroon" title="c5">a</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D1" title="GROUP_1:def.1">GROUP_1:def 1</a></i>
;<br/><br/>


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