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<div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="binop_1.html#NM2">BinOp</a> of  <a href="midsp_1.html#K13" target="_self">setvect</a> <font color="Maroon">M</font>, <font color="Maroon">g</font> be   <a href="binop_1.html#NM2">BinOp</a> of  <a href="midsp_1.html#K13" target="_self">setvect</a> <font color="Maroon">M</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:66_1_2"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">u1</font>, <font color="Olive">v1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="midsp_1.html#K13" target="_self">setvect</a> <font color="Maroon">M</font> holds  <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">u1</font>,<font color="Olive">v1</font> <a href="hidden.html#R1">=</a> <font color="Olive">u1</font> <a href="midsp_1.html#K14" target="_self">+</a> <font color="Olive">v1</font>
 <b>and </b><br/><a NAME="E2:66_1_2"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">u1</font>, <font color="Olive">v1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="midsp_1.html#K13" target="_self">setvect</a> <font color="Maroon">M</font> holds  <font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Olive">u1</font>,<font color="Olive">v1</font> <a href="hidden.html#R1">=</a> <font color="Olive">u1</font> <a href="midsp_1.html#K14" target="_self">+</a> <font color="Olive">v1</font>
 ;<br/>

<a NAME="E3:66_1_2"/>
for <font color="Olive">u1</font>, <font color="Olive">v1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="midsp_1.html#K13" target="_self">setvect</a> <font color="Maroon">M</font> holds  <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">u1</font>,<font color="Olive">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Olive">u1</font>,<font color="Olive">v1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="midsp_1.html#K13" target="_self">setvect</a> <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">v1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="midsp_1.html#K13" target="_self">setvect</a> <font color="Maroon">M</font>;<br/>



<a NAME="E1:66_1_2_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font> <a href="midsp_1.html#K14" target="_self">+</a> <font color="Maroon">v1</font>
 
<b>by </b><i><a class="ref" href="midsp_1.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E2:66_1_2_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">v1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font>
 
<b>by </b><i><a class="ref" href="midsp_1.html#T29" target="_self">Th29</a>, </i>;<br/>
<b>hence </b><a NAME="E3:66_1_2_1"/>
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:66_1_2"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


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