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<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">J</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>, <font color="Maroon">G</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">I</font>,<font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">J</font>,<font color="Maroon">D</font>;<br/>









<b>assume </b><a NAME="E1:37"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">F</font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">F</font> is <a href="binop_1.html#V2">associative</a> &amp; <font color="Maroon">F</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  &amp; <font color="Maroon">G</font> is <a href="binop_1.html#V1">commutative</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font>;<br/><b>thus </b><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">y</font></span><a href="setwiseo.html#K2">.}</a></span></span><a href="matrix_3.html#K8" target="_self">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> = 
<font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">y</font></span><a href="setwiseo.html#K2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2">.}</a></span></span><a href="matrix_3.html#K8" target="_self">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">g</font>,<font color="Maroon">f</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">y</font></span><a href="setwiseo.html#K2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p3"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2">.}</a></span>,<font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">y</font></span><a href="setwiseo.html#K2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funcop_1.html#K8">[;]</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p3"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2">.}</a></span>,<font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T79">FUNCOP_1:79</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:37"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font><br/> for <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> holds  <font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Olive">x</font></span><a href="setwiseo.html#K2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Olive">y</font></span><a href="setwiseo.html#K2">.}</a></span></span><a href="matrix_3.html#K8" target="_self">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Olive">y</font></span><a href="setwiseo.html#K2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funcop_1.html#K8">[;]</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p3"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Olive">x</font></span><a href="setwiseo.html#K2">.}</a></span>,<font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span>
 ;<br/>


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