<?xml version="1.0"?>
<div class="add">

<span class="kw">set </span><font color="Maroon" title="c4">X</font> =  <a href="csspace.html#K9" title="CSSPACE:func.9">the_set_of_l2ComplexSequences</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">scalar1</font>, <font color="Maroon" title="c6">scalar2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="csspace.html#K9" title="CSSPACE:func.9">the_set_of_l2ComplexSequences</a>,<a href="csspace.html#K9" title="CSSPACE:func.9">the_set_of_l2ComplexSequences</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace.html#K9" title="CSSPACE:func.9">the_set_of_l2ComplexSequences</a>  &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace.html#K9" title="CSSPACE:func.9">the_set_of_l2ComplexSequences</a>  holds <br/><font color="Maroon" title="c5">scalar1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_3.html#K11" title="COMSEQ_3:func.11">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">*'</a></span>)</span></span>)</span> ) &amp; (  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace.html#K9" title="CSSPACE:func.9">the_set_of_l2ComplexSequences</a>  &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace.html#K9" title="CSSPACE:func.9">the_set_of_l2ComplexSequences</a>  holds <br/><font color="Maroon" title="c6">scalar2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_3.html#K11" title="COMSEQ_3:func.11">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">*'</a></span>)</span></span>)</span> ) implies <font color="Maroon" title="c5">scalar1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">scalar2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:105_1_2"/><span class="lab"><font color="Green" title="E50">A2</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace.html#K9" title="CSSPACE:func.9">the_set_of_l2ComplexSequences</a>  &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace.html#K9" title="CSSPACE:func.9">the_set_of_l2ComplexSequences</a>  holds <br/><font color="Maroon" title="c5">scalar1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_3.html#K11" title="COMSEQ_3:func.11">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">*'</a></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:105_1_2"/><span class="lab"><font color="Green" title="E51">A3</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace.html#K9" title="CSSPACE:func.9">the_set_of_l2ComplexSequences</a>  &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace.html#K9" title="CSSPACE:func.9">the_set_of_l2ComplexSequences</a>  holds <br/><font color="Maroon" title="c6">scalar2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_3.html#K11" title="COMSEQ_3:func.11">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">*'</a></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">scalar1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">scalar2</font></span><br/>

<a NAME="E3:105_1_2"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="csspace.html#K9" title="CSSPACE:func.9">the_set_of_l2ComplexSequences</a>  &amp; <font color="Olive" title="b2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="csspace.html#K9" title="CSSPACE:func.9">the_set_of_l2ComplexSequences</a>  holds <br/><font color="Maroon" title="c5">scalar1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">scalar2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="105_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="csspace.html#K9" title="CSSPACE:func.9">the_set_of_l2ComplexSequences</a>  &amp; <font color="Maroon" title="c8">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="csspace.html#K9" title="CSSPACE:func.9">the_set_of_l2ComplexSequences</a>  implies <font color="Maroon" title="c5">scalar1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">scalar2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:105_1_2_1"/><span class="lab"><font color="Green" title="E52">A4</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="csspace.html#K9" title="CSSPACE:func.9">the_set_of_l2ComplexSequences</a> 
 <span class="kw">and </span><br/><a NAME="E2:105_1_2_1"/><span class="lab"><font color="Green" title="E53">A5</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="csspace.html#K9" title="CSSPACE:func.9">the_set_of_l2ComplexSequences</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">scalar1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">scalar2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>)</span><br/>

<span class="kw">thus </span><font color="Maroon" title="c5">scalar1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>) = 
 <a href="comseq_3.html#K11" title="COMSEQ_3:func.11">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">*'</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105_1_2"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="txt" href="#E1:105_1_2_1"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E2:105_1_2_1"><span class="lab"><font color="Green" title="E53">A5</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">scalar2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:105_1_2"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="txt" href="#E1:105_1_2_1"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E2:105_1_2_1"><span class="lab"><font color="Green" title="E53">A5</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:105_1_2"/>
<font color="Maroon" title="c5">scalar1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">scalar2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
