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<b>let </b><font color="Maroon" title="c6">o1</font>, <font color="Maroon" title="c7">o2</font> be   <a href="monoid_1.html#NM1" title="MONOID_1:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">D1</font></span>)</span>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c2">D2</font></span>)</span>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c3">D</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:12_1_2"/><i><font color="Green" title="E2">A2</font></i>: 
for <font color="Olive" title="b1">f1</font> being   <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="c5">A</font>,<font color="Maroon" title="c1">D1</font><br/> for <font color="Olive" title="b2">f2</font> being   <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="c5">A</font>,<font color="Maroon" title="c2">D2</font> holds  <font color="Maroon" title="c6">o1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="monoid_1.html#K3" title="MONOID_1:func.3">.:</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>
 <b>and </b><br/><a NAME="E2:12_1_2"/><i><font color="Green" title="E3">A3</font></i>: 
for <font color="Olive" title="b1">f1</font> being   <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="c5">A</font>,<font color="Maroon" title="c1">D1</font><br/> for <font color="Olive" title="b2">f2</font> being   <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="c5">A</font>,<font color="Maroon" title="c2">D2</font> holds  <font color="Maroon" title="c7">o2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="monoid_1.html#K3" title="MONOID_1:func.3">.:</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">f1</font> be    <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="c5">A</font>,<font color="Maroon" title="c1">D1</font>;<br/><b>let </b><font color="Maroon" title="c9">f2</font> be    <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="c5">A</font>,<font color="Maroon" title="c2">D2</font>;<br/><b>thus </b><font color="Maroon" title="c6">o1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c9">f2</font> = 
<font color="Maroon" title="c4">f</font> <a href="monoid_1.html#K3" title="MONOID_1:func.3">.:</a> <font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c9">f2</font>
<b>by </b><i><a class="txt" href="monoid_1.html#E1:12_1_2"><i><font color="Green" title="E2">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c7">o2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c9">f2</font>
<b>by </b><i><a class="txt" href="monoid_1.html#E2:12_1_2"><i><font color="Green" title="E3">A3</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:12_1_2"/>
<font color="Maroon" title="c6">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">o2</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></i>;<br/>


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