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

<span class="kw">let </span><font color="Maroon" title="c1">cF</font> be   <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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c1">cF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">Rev</a> <font color="Maroon" title="c1">cF</font></span>)</span></span><br/>

<a NAME="E1:104"/>
<font color="Maroon" title="c1">cF</font> is  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="afinsq_2.html#E58"><span class="lab"><font color="Green" title="E47">Lm3</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">Fr2</font> = <font color="Maroon" title="c1">cF</font>, <font color="Maroon" title="c3">Fr1</font> =  <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">Rev</a> <font color="Maroon" title="c1">cF</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  ;<br/>
<a NAME="E3:104"/><span class="lab"><font color="Green" title="E62">A1</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">Fr2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#D1" target="_self" title="AFINSQ_2:def.1">Def1</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">i</font></span>)</span>;<br/>
<a NAME="E4:104"/><span class="lab"><font color="Green" title="E63">A2</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font></span>)</span>
 
;<br/>
<a NAME="E5:104"/><span class="lab"><font color="Green" title="E64">A3</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font> holds <br/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font> implies  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:104_1"/><span class="lab"><font color="Green" title="E65">A4</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">k</font> = <font color="Maroon" title="c4">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T1" target="_self" title="AFINSQ_2:th.1">Th1</a>, <a class="txt" href="#E1:104_1"><span class="lab"><font color="Green" title="E65">A4</font></span></a></span>;<br/>
<a NAME="E3:104_1"/>
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104_1"><span class="lab"><font color="Green" title="E65">A4</font></span></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/>
<a NAME="E4:104_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E5:104_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A5</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c5">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/>
<span class="kw">take </span>
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c5">k</font></span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c5">k</font></span>)</span>] )</span><br/>

<a NAME="E6:104_1"/>
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c5">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E7:104_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c5">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:104_1"/>
( <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c5">k</font></span>)</span>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:104_1"><span class="lab"><font color="Green" title="E66">A5</font></span></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>; <a class="txt" 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">consider </span><font color="Maroon" title="c4">P</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font></span>)</span>,<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E7:104"/><span class="lab"><font color="Green" title="E65">A6</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">P</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="#E5:104"><span class="lab"><font color="Green" title="E64">A3</font></span></a>);<br/></span>
<a NAME="E8:104"/><span class="lab"><font color="Green" title="E66">A7</font></span>: 
 for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font> &amp; <font color="Maroon" title="c4">P</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font> &amp; <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font> &amp; <font color="Maroon" title="c4">P</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">x2</font> implies <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:104_2"/><span class="lab"><font color="Green" title="E67">A8</font></span>: 
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font>
 <span class="kw">and </span><br/><a NAME="E2:104_2"/><span class="lab"><font color="Green" title="E68">A9</font></span>: 
<font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font>
 <span class="kw">and </span><br/><a NAME="E3:104_2"/><span class="lab"><font color="Green" title="E69">A10</font></span>: 
<font color="Maroon" title="c4">P</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">i</font> = <font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c8">j</font> = <font color="Maroon" title="c6">x2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104_2"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="txt" href="#E2:104_2"><span class="lab"><font color="Green" title="E68">A9</font></span></a>, <a class="ref" href="afinsq_2.html#T1" target="_self" title="AFINSQ_2:th.1">Th1</a></span>;<br/>
<a NAME="E5:104_2"/><span class="lab"><font color="Green" title="E70">A11</font></span>: 
<font color="Maroon" title="c4">P</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c8">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:104"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="txt" href="#E2:104_2"><span class="lab"><font color="Green" title="E68">A9</font></span></a></span>;<br/>
<a NAME="E6:104_2"/>
<font color="Maroon" title="c4">P</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c7">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:104"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="txt" href="#E1:104_2"><span class="lab"><font color="Green" title="E67">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:104_2"/>
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104_2"><span class="lab"><font color="Green" title="E69">A10</font></span></a>, <a class="txt" href="#E5:104_2"><span class="lab"><font color="Green" title="E70">A11</font></span></a></span>; <a class="txt" 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="E9:104"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A12</font></span>: 
<font color="Maroon" title="c4">P</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T56" title="FUNCT_2:th.56">FUNCT_2:56</a></span>;<br/>
<a NAME="E10:104"/>
<font color="Maroon" title="c4">P</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:104"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="ref" href="funct_2.html#T56" title="FUNCT_2:th.56">FUNCT_2:56</a></span>;<br/>
<a NAME="E11:104"/><span class="kw">then </span>
<font color="Maroon" title="c4">P</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:104"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="txt" href="afinsq_2.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">P</font> = <font color="Maroon" title="c4">P</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">Fr1</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:104"><span class="lab"><font color="Green" title="E67">A12</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E68">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="104_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">Fr1</font> implies <font color="Maroon" title="c3">Fr1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Fr2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:104_3"/><span class="lab"><font color="Green" title="E69">A14</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">Fr1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">Fr1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Fr2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c7">k</font> = <font color="Maroon" title="c6">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104_3"><span class="lab"><font color="Green" title="E69">A14</font></span></a></span>;<br/><a NAME="E3:104_3"/>
<font color="Maroon" title="c5">P</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">Fr1</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c7">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:104"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="txt" href="#E1:104_3"><span class="lab"><font color="Green" title="E69">A14</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:104_3"/>
<font color="Maroon" title="c3">Fr1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Fr2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="ref" href="afinsq_2.html#D1" target="_self" title="AFINSQ_2:def.1">Def1</a>, <a class="txt" href="#E1:104_3"><span class="lab"><font color="Green" title="E69">A14</font></span></a></span>; <a class="txt" 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>
<div><span class="lab"><font color="Green" title="E69">A15</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="104_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">Fr1</font> implies ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c5">P</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">Fr2</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:104_4"/><span class="lab"><font color="Green" title="E70">A16</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">Fr1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c5">P</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">Fr2</font> )</span><br/><a NAME="E2:104_4"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104_4"><span class="lab"><font color="Green" title="E70">A16</font></span></a>, <a class="ref" href="funct_2.html#T52" title="FUNCT_2:th.52">FUNCT_2:52</a></span>;<br/><a NAME="E3:104_4"/><span class="kw">then </span>
<font color="Maroon" title="c5">P</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:104_4"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c5">P</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">Fr2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E1:104_4"><span class="lab"><font color="Green" title="E70">A16</font></span></a>, <a class="ref" href="funct_2.html#T52" title="FUNCT_2:th.52">FUNCT_2:52</a></span>; <a class="txt" 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="E15:104"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c5">P</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">Fr2</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">Fr1</font>
 
;<br/>
<a NAME="E16:104"/><span class="kw">then </span>
<font color="Maroon" title="c3">Fr1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Fr2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:104"><span class="lab"><font color="Green" title="E69">A15</font></span></a>, <a class="txt" href="#E13:104"><span class="lab"><font color="Green" title="E68">A13</font></span></a>, <a class="ref" href="funct_1.html#T10" title="FUNCT_1:th.10">FUNCT_1:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:104"/>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c1">cF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">Rev</a> <font color="Maroon" title="c1">cF</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="ref" href="afinsq_2.html#T45" target="_self" title="AFINSQ_2:th.45">Th57</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
