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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">f1</font>, <font color="Olive" title="b2">f2</font> being  <font color="Maroon" title="c1">n</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>   <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a> holds   <a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="valued_1.html#K18" title="VALUED_1:func.18">(#)</a> <font color="Olive" title="b2">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Olive" title="b1">f1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Olive" title="b2">f2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</font> be  <font color="Maroon" title="c1">n</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>   <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</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">  <a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K18" title="VALUED_1:func.18">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c3">f2</font></span>)</span></span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being  $1 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>   <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a> holds   <a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="valued_1.html#K18" title="VALUED_1:func.18">(#)</a> <font color="Olive" title="b2">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Olive" title="b1">f1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Olive" title="b2">f2</font></span>)</span>;<br/>
<a NAME="E1:111"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="111_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:111_1"/><span class="lab"><font color="Green" title="E41">B1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</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"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:111_1"/>
 for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being  <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>   <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a> holds   <a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="valued_1.html#K18" title="VALUED_1:func.18">(#)</a> <font color="Olive" title="b2">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Olive" title="b1">f1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Olive" title="b2">f2</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="111_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">f1</font>, <font color="Maroon" title="c6">f2</font> be  <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>   <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</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">  <a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="valued_1.html#K18" title="VALUED_1:func.18">(#)</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c6">f2</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">F</font> = <font color="Maroon" title="c5">f1</font> <a href="valued_1.html#K18" title="VALUED_1:func.18">(#)</a> <font color="Maroon" title="c6">f2</font> as  <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>   <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">G</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="valued_1.html#K18" title="VALUED_1:func.18">(#)</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c4">k</font> as  <font color="Maroon" title="c4">k</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>   <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">g1</font> = <font color="Maroon" title="c5">f1</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c4">k</font> as  <font color="Maroon" title="c4">k</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>   <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">g2</font> = <font color="Maroon" title="c6">f2</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c4">k</font> as  <font color="Maroon" title="c4">k</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>   <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a> ;<br/>
<a NAME="E5:111_1_1"/><span class="lab"><font color="Green" title="E42">C1</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<a NAME="E6:111_1_1"/>
<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E7:111_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">C2</font></span>: 
( <font color="Maroon" title="c4">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">f1</font></span>)</span> &amp; <font color="Maroon" title="c4">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">f2</font></span>)</span> &amp; <font color="Maroon" title="c4">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">F</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:111_1_1"><span class="lab"><font color="Green" title="E42">C1</font></span></a>, <a class="ref" href="nat_1.html#T44" onmouseover="rs('nat_1/T44')" onmouseout="rh()">NAT_1:44</a></span>;<br/>
<a NAME="E8:111_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">C3</font></span>: 
(  <a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c9">g1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> &amp;  <a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c10">g2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_4.html#T52" target="_self" onmouseover="rs('rvsum_4/T52')" onmouseout="rh()">A265</a></span>;<br/>
<a NAME="E9:111_1_1"/><span class="lab"><font color="Green" title="E45">C4</font></span>: 
 <a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="valued_1.html#K18" title="VALUED_1:func.18">(#)</a> <font color="Maroon" title="c10">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c9">g1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c10">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_1"><span class="lab"><font color="Green" title="E41">B1</font></span></a></span>;<br/>
 <a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c7">F</font> = 
 <a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c8">G</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:111_1_1"><span class="lab"><font color="Green" title="E43">C2</font></span></a>, <a class="ref" href="rvsum_4.html#T52" target="_self" onmouseover="rs('rvsum_4/T52')" onmouseout="rh()">A265</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c9">g1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c10">g2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="valued_1.html#K18" title="VALUED_1:func.18">(#)</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:111_1_1"><span class="lab"><font color="Green" title="E45">C4</font></span></a>, <a class="ref" href="rvsum_4.html#T3" target="_self" onmouseover="rs('rvsum_4/T3')" onmouseout="rh()">MFG</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c9">g1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c10">g2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:111_1_1"><span class="lab"><font color="Green" title="E43">C2</font></span></a>, <a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c6">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:111_1_1"><span class="lab"><font color="Green" title="E44">C3</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E11:111_1_1"/>
 <a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="valued_1.html#K18" title="VALUED_1:func.18">(#)</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c6">f2</font></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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:111_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <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>
<a NAME="E2:111"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="111_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">f2</font> be   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>   <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</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">  <a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="valued_1.html#K18" title="VALUED_1:func.18">(#)</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c5">f2</font></span>)</span></span><br/>

<a NAME="E1:111_2"/>
(  <a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="valued_1.html#NK26" title="VALUED_1:NK.26">(#)</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c5">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_4.html#T51" target="_self" onmouseover="rs('rvsum_4/T51')" onmouseout="rh()">PFO</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:111_2"/>
 <a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="valued_1.html#K18" title="VALUED_1:func.18">(#)</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c5">f2</font></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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:111"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E2:111"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E1:111"><span class="lab"><font color="Green" title="E40">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:111"/>
 <a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K18" title="VALUED_1:func.18">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K14" title="RVSUM_4:func.14">XProduct</a> <font color="Maroon" title="c3">f2</font></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"> verum</span><br/>


</div>
