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<div class="add">

<b>set </b><font color="Maroon">X</font> =  <a href="csspace.html#K11" target="_self">the_set_of_l2ComplexSequences</a> ;<br/>
<b>let </b><font color="Maroon">scalar1</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="csspace.html#K11" target="_self">the_set_of_l2ComplexSequences</a> ,<a href="csspace.html#K11" target="_self">the_set_of_l2ComplexSequences</a> </span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K2">COMPLEX</a> , <font color="Maroon">scalar2</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="csspace.html#K11" target="_self">the_set_of_l2ComplexSequences</a> ,<a href="csspace.html#K11" target="_self">the_set_of_l2ComplexSequences</a> </span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K2">COMPLEX</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:112_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="csspace.html#K11" target="_self">the_set_of_l2ComplexSequences</a>  &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="csspace.html#K11" target="_self">the_set_of_l2ComplexSequences</a>  holds <br/><font color="Maroon">scalar1</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="comseq_3.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Olive">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Olive">y</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:112_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="csspace.html#K11" target="_self">the_set_of_l2ComplexSequences</a>  &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="csspace.html#K11" target="_self">the_set_of_l2ComplexSequences</a>  holds <br/><font color="Maroon">scalar2</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="comseq_3.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Olive">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Olive">y</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E3:112_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="csspace.html#K11" target="_self">the_set_of_l2ComplexSequences</a>  &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="csspace.html#K11" target="_self">the_set_of_l2ComplexSequences</a>  holds <br/><font color="Maroon">scalar1</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">scalar2</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:112_1_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="csspace.html#K11" target="_self">the_set_of_l2ComplexSequences</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="csspace.html#K11" target="_self">the_set_of_l2ComplexSequences</a>  )
 ;<br/>

<b>thus </b><font color="Maroon">scalar1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
 <a href="comseq_3.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="comseq_1.html#K3">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">y</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#T8" target="_self">Th8</a>, <a class="ref" href="csspace.html#T10" target="_self">Th10</a></i>
<br/>.= 
<font color="Maroon">scalar2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="csspace.html#T9" target="_self">Th9</a>, <a class="ref" href="csspace.html#T10" target="_self">Th10</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:112_1_1"/>
<font color="Maroon">scalar1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">scalar2</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T1">BINOP_1:1</a></i>;<br/>


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