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<b>let </b><font color="Maroon">f1</font> be   <a href="binop_1.html#NM2">BinOp</a> of  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">V</font>,<font color="Maroon">W</font>, <font color="Maroon">f2</font> be   <a href="binop_1.html#NM2">BinOp</a> of  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">V</font>,<font color="Maroon">W</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:34_1_2"/><i><font color="Green">E104</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">V</font>,<font color="Maroon">W</font><br/> for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>  st <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font> holds <br/><font color="Maroon">f1</font> <a href="binop_1.html#K2">.</a> <font color="Olive">A</font>,<font color="Olive">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b</font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E2:34_1_2"/><i><font color="Green">E105</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">V</font>,<font color="Maroon">W</font><br/> for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>  st <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font> holds <br/><font color="Maroon">f2</font> <a href="binop_1.html#K2">.</a> <font color="Olive">A</font>,<font color="Olive">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b</font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font>
 ;<br/>

<b>set </b><font color="Maroon">C</font> =  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">V</font>,<font color="Maroon">W</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">V</font>,<font color="Maroon">W</font>, <font color="Maroon">B</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">V</font>,<font color="Maroon">W</font>;<br/><a NAME="E1:34_1_2_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">V</font>,<font color="Maroon">W</font>
 ;<br/><b>then consider </b><font color="Maroon">A1</font> being    <a href="vectsp_4.html#M2">Coset</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E3:34_1_2_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E4:34_1_2_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">V</font>,<font color="Maroon">W</font>
 ;<br/><b>then consider </b><font color="Maroon">B1</font> being    <a href="vectsp_4.html#M2">Coset</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E6:34_1_2_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 ;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E8:34_1_2_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D6">VECTSP_4:def 6</a></i>;<br/><b>consider </b><font color="Maroon">b</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E10:34_1_2_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D6">VECTSP_4:def 6</a></i>;<br/><b>thus </b><font color="Maroon">f1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font>
<b>by </b><i>, , , , </i>
<br/>.= 
<font color="Maroon">f2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
<b>by </b><i>, , , , </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:34_1_2"/>
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


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