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

<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">L1</font> be   <a href="convex2.html#NM1">Convex_Combination</a> of <font color="Maroon">V</font>, <font color="Maroon">L2</font> be   <a href="convex2.html#NM1">Convex_Combination</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><a NAME="E1:22"/><i><font color="Green">E29</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 ;<br/>

<a NAME="E2:22"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E3:22"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/>
<b>consider </b><font color="Maroon">F1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E5:22"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">F1</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font> &amp;  ex <font color="Olive">f</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> &amp;  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">f</font> holds <br/>( <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) ) ) )
 <b>by </b><i><a class="ref" href="convex1.html#D3">CONVEX1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">f1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E7:22"/><i><font color="Green">E41</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> &amp;  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font> holds <br/>( <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">F2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E9:22"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">F2</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L2</font> &amp;  ex <font color="Olive">f</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font> &amp;  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">f</font> holds <br/>( <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) ) ) )
 <b>by </b><i><a class="ref" href="convex1.html#D3">CONVEX1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">f2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E11:22"/><i><font color="Green">E44</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font> &amp;  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font> holds <br/>( <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) ) )
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">L</font> = <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span>;<br/>
<a NAME="E12:22"/><i><font color="Green">E45</font></i>: 
(  <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font> &amp;  <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L2</font> )
 
<b>by </b><i>, , <a class="ref" href="rlvect_2.html#T65">RLVECT_2:65</a></i>;<br/>
<a NAME="E13:22"/><i><font color="Green">E47</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">Top</font> = <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">Mid</font> = <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">Btm</font> = <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span></span>)</span>;<br/>
<a NAME="E14:22"/>
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<b>then consider </b><font color="Maroon">Lt</font> being    <a href="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E16:22"/><i><font color="Green">E48</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">Lt</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E17:22"/>
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T29">XBOOLE_1:29</a></i>;<br/>
<b>then consider </b><font color="Maroon">Lm</font> being    <a href="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E19:22"/><i><font color="Green">E49</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">Lm</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E20:22"/>
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<b>then consider </b><font color="Maroon">Lb</font> being    <a href="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E22:22"/><i><font color="Green">E50</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">Lb</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">Ft</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E24:22"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">Ft</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ft</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">Lt</font> &amp;  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">Lt</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Lt</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">Ft</font></span>)</span> )
 <b>by </b><i><a class="ref" href="rlvect_2.html#D10">RLVECT_2:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">Fm</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E26:22"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">Fm</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fm</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">Lm</font> &amp;  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">Lm</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Lm</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">Fm</font></span>)</span> )
 <b>by </b><i><a class="ref" href="rlvect_2.html#D10">RLVECT_2:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">Fb</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E28:22"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">Fb</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fb</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">Lb</font> &amp;  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">Lb</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Lb</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">Fb</font></span>)</span> )
 <b>by </b><i><a class="ref" href="rlvect_2.html#D10">RLVECT_2:def 10</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">ft</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E30:22"/><i><font color="Green">E60</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">ft</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Ft</font> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Ft</font></span>)</span> holds <br/><font color="Maroon">ft</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">j</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E31:22"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">ft</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ft</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ft</font> &amp; <font color="Maroon">ft</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E5:22_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Ft</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:22_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Ft</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:22_1"/><b>then </b>
( <font color="Maroon">Ft</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ft</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ft</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Ftx</font> = <font color="Maroon">Ft</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> ;<br/>
<a NAME="E9:22_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">ft</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="convex2.html#E1:22"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">L1f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E11:22_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1f</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E12:22_1"/>
<font color="Maroon">Ftx</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1f</font>
 
<b>by </b><i/>;<br/>
<a NAME="E13:22_1"/><b>then </b>
<font color="Maroon">ft</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1f</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>hence </b><a NAME="E14:22_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">ft</font> = <font color="Maroon">ft</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">fm1</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E34:22"/><i><font color="Green">E66</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fm1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Fm</font> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Fm</font></span>)</span> holds <br/><font color="Maroon">fm1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">j</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E35:22"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">fm1</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">fm1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fm1</font> &amp; <font color="Maroon">fm1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="convex2.html#E1:22"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E5:22_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Fm</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="convex2.html#E1:22"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:22_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fm</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:22_2"/><b>then </b>
( <font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fm</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fm</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Fmx</font> = <font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> ;<br/>
<a NAME="E9:22_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">fm1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">L1f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E11:22_2"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1f</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E12:22_2"/>
<font color="Maroon">Fmx</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1f</font>
 
<b>by </b><i/>;<br/>
<a NAME="E13:22_2"/><b>then </b>
<font color="Maroon">fm1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1f</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>hence </b><a NAME="E14:22_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="txt" href="convex2.html#E1:22"><i><font color="Green">E29</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">fm1</font> = <font color="Maroon">fm1</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">fm2</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E38:22"/><i><font color="Green">E72</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fm2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Fm</font> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Fm</font></span>)</span> holds <br/><font color="Maroon">fm2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">j</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E39:22"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">fm2</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">fm2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_3"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fm2</font> &amp; <font color="Maroon">fm2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E5:22_3"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Fm</font></span>)</span>
 
<b>by </b><i><a class="txt" href="convex2.html#E1:22"><i><font color="Green">E29</font></i></a>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:22_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fm</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:22_3"/><b>then </b>
( <font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fm</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fm</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Fmx</font> = <font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> ;<br/>
<a NAME="E9:22_3"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">fm2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="convex2.html#E1:22"><i><font color="Green">E29</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">L2f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E11:22_3"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2f</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E12:22_3"/>
<font color="Maroon">Fmx</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2f</font>
 
<b>by </b><i/>;<br/>
<a NAME="E13:22_3"/><b>then </b>
<font color="Maroon">fm2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2f</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>hence </b><a NAME="E14:22_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">fm2</font> = <font color="Maroon">fm2</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">fb</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E42:22"/><i><font color="Green">E77</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fb</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Fb</font> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Fb</font></span>)</span> holds <br/><font color="Maroon">fb</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">j</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E43:22"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">fb</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">fb</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_4"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fb</font> &amp; <font color="Maroon">fb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E5:22_4"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Fb</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:22_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fb</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:22_4"/><b>then </b>
( <font color="Maroon">Fb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fb</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fb</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Fbx</font> = <font color="Maroon">Fb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> ;<br/>
<a NAME="E9:22_4"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">fb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">L2f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E11:22_4"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2f</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E12:22_4"/>
<font color="Maroon">Fbx</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2f</font>
 
<b>by </b><i/>;<br/>
<a NAME="E13:22_4"/><b>then </b>
<font color="Maroon">fb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2f</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>hence </b><a NAME="E14:22_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">fb</font> = <font color="Maroon">fb</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E45:22"/>
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E46:22"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ft</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fm</font>
 
<b>by </b><i>, , , , <a class="ref" href="xboole_1.html#T85">XBOOLE_1:85</a></i>;<br/>
<a NAME="E47:22"/>
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E48:22"/><b>then </b><i><font color="Green">E83</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fm</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fb</font>
 
<b>by </b><i>, , , , <a class="ref" href="xboole_1.html#T85">XBOOLE_1:85</a></i>;<br/>
<a NAME="E49:22"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">F1</font>,<font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_5"><b>proof </b></a><div class="add">

<a NAME="E1:22_5"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, , , <a class="ref" href="finseq_3.html#T98">FINSEQ_3:98</a></i>;<br/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">Ft</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">Fm</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span>
<b>by </b><i>, , , , , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T42">XBOOLE_1:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T52">XBOOLE_1:52</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font></span>)</span> <a href="subset_1.html#K7">\</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>
<br/>.= 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E3:22_5"/>
<font color="Maroon">F1</font>,<font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i>, , <a class="ref" href="rfinseq.html#T39">RFINSEQ:39</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E50:22"/>
<font color="Maroon">f1</font>,<font color="Maroon">ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fm1</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6"><b>proof </b></a><div class="add">

<a NAME="E1:22_6"/><i><font color="Green">E86</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T169">FUNCT_2:169</a></i>;<br/>
<a NAME="E2:22_6"/><i><font color="Green">E87</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_1"><b>proof </b></a><div class="add">

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

<div><i><font color="Green">E88</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_6_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font>
 ;<br/><a NAME="E2:22_6_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E3:22_6_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:22_6_1_1"/><b>then </b>
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E5:22_6_1_1"/>
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_6_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font> )
 ;<br/><a NAME="E2:22_6_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E3:22_6_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:22_6_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font> iff ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:22_6"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font> holds <br/><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:22_6_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:22_6_2"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:22_6_2"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:22_6"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">F1</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T20">FUNCT_1:20</a></i>;<br/>
<a NAME="E5:22_6"/><i><font color="Green">E91</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">Lt</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">Lm</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E6:22_6"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fm1</font></span>)</span> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_3"><b>proof </b></a><div class="add">

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

<i><font color="Green">E93</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fm1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ft</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/>
<i><font color="Green">E94</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fm1</font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fm1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E3:22_6_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fm1</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_3_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_6_3_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fm1</font></span>)</span>
 ;<br/>

<a NAME="E2:22_6_3_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span>
 
<b>by </b><i><a class="txt" href="convex2.html#E5:22"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E3:22_6_3_3"/><b>then </b><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">Lt</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">Lm</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E4:22_6_3_3"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T169">FUNCT_2:169</a></i>;<br/>
<b>hence </b><a NAME="E5:22_6_3_3"/>
<span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font>
 <b>by </b><i><a class="txt" href="convex2.html#E7:22"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:22_6_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fm1</font></span>)</span> iff ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font> ) )
 <b>by </b><i><a class="txt" href="convex2.html#E5:22"><i><font color="Green">E40</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:22_6"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fm1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fm1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:22_6_4"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fm1</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_6_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_6_4_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ft</font> or  ex <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fm1</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ft</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">m</font> ) )
 </span><b>by </b><i>, <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_6_4_1_1_1"><b>suppose </b></a><a NAME="E1:22_6_4_1_1_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ft</font>
 ;<br/><div class="add"><a NAME="E2:22_6_4_1_1_1"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Ft</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:22_6_4_1_1_1"/><b>then </b>
<font color="Maroon">ft</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:22_6_4_1_1_1"/><b>then </b><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fm1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="convex2.html#E5:22"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E5:22_6_4_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Ft</font>
 <b>by </b><i><a class="txt" href="convex2.html#E7:22"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E6:22_6_4_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fm1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_6_4_1_1_2"><b>suppose </b></a><a NAME="E1:22_6_4_1_1_2"/>
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fm1</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ft</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">m</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:22_6_4_1_1_2"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fm1</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ft</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> )
 ;<br/><a NAME="E4:22_6_4_1_1_2"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Fm</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="convex2.html#E5:22"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><i><font color="Green">E102</font></i>: <span class="p1">(<span class="default"><font color="Maroon">ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fm1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">fm1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="txt" href="convex2.html#E5:22"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i>, <a class="txt" href="convex2.html#E7:22"><i><font color="Green">E41</font></i></a></i>
;<br/><a NAME="E6:22_6_4_1_1_2"/>
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fm</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Ft</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> )
 <b>by </b><i>, <a class="txt" href="convex2.html#E5:22"><i><font color="Green">E40</font></i></a>, <a class="txt" href="convex2.html#E7:22"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E7:22_6_4_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fm1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:22_6_4"/>
<span class="p1">(<span class="default"><font color="Maroon">ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fm1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:22_6"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fm1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span>
 
<b>by </b><i><a class="txt" href="convex2.html#E3:22"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#T20">FUNCT_1:20</a></i>;<br/>
<a NAME="E9:22_6"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rfinseq.html#T16">RFINSEQ:16</a></i>;<br/>
<b>hence </b><a NAME="E10:22_6"/>
<font color="Maroon">f1</font>,<font color="Maroon">ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fm1</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i>, , , , <a class="txt" href="convex2.html#E2:22"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="bagorder.html#T4">BAGORDER:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E51:22"/><b>then </b><i><font color="Green">E104</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fm1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfinseq.html#T22">RFINSEQ:22</a></i>;<br/>
<a NAME="E52:22"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">F2</font>,<font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_7"><b>proof </b></a><div class="add">

<a NAME="E1:22_7"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, , , <a class="ref" href="finseq_3.html#T98">FINSEQ_3:98</a></i>;<br/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T42">XBOOLE_1:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T52">XBOOLE_1:52</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L2</font></span>)</span> <a href="subset_1.html#K7">\</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>
<br/>.= 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F2</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E3:22_7"/>
<font color="Maroon">F2</font>,<font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i>, , <a class="ref" href="rfinseq.html#T39">RFINSEQ:39</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E53:22"/>
<font color="Maroon">f2</font>,<font color="Maroon">fm2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fb</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_8"><b>proof </b></a><div class="add">

<a NAME="E1:22_8"/><i><font color="Green">E107</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T169">FUNCT_2:169</a></i>;<br/>
<a NAME="E2:22_8"/><i><font color="Green">E108</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_8_1"><b>proof </b></a><div class="add">

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

<div><i><font color="Green">E109</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_8_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_8_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font>
 ;<br/><a NAME="E2:22_8_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E3:22_8_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:22_8_1_1"/><b>then </b>
<font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F2</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E5:22_8_1_1"/>
<font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_8_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_8_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2</font> )
 ;<br/><a NAME="E2:22_8_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E3:22_8_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:22_8_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font> iff ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2</font> ) )
 <b>by </b><i><a class="txt" href="convex2.html#E3:22"><i><font color="Green">E38</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:22_8"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font> holds <br/><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_8_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:22_8_2"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:22_8_2"/>
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="convex2.html#E3:22"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E4:22_8_2"/>
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:22_8"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">F2</font>
 
<b>by </b><i><a class="txt" href="convex2.html#E2:22"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#T20">FUNCT_1:20</a></i>;<br/>
<a NAME="E5:22_8"/><i><font color="Green">E112</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">Lm</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">Lb</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E6:22_8"/><i><font color="Green">E113</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fm2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fb</font></span>)</span> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_8_3"><b>proof </b></a><div class="add">

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

<i><font color="Green">E118</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">fm2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fb</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fm2</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fb</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span>
<b>by </b><i><a class="txt" href="convex2.html#E1:22"><i><font color="Green">E29</font></i></a>, , <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/>
<i><font color="Green">E119</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fm2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fb</font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">fm2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fb</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span>
<b>by </b><i><a class="txt" href="convex2.html#E7:22"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E3:22_8_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fm2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fb</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_8_3_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_8_3_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fm2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fb</font></span>)</span>
 ;<br/>

<a NAME="E2:22_8_3_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E3:22_8_3_3"/><b>then </b><i><font color="Green">E120</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">Lm</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">Lb</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E4:22_8_3_3"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T169">FUNCT_2:169</a></i>;<br/>
<b>hence </b><a NAME="E5:22_8_3_3"/>
<span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2</font>
 <b>by </b><i><a class="txt" href="convex2.html#E9:22"><i><font color="Green">E43</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:22_8_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fm2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fb</font></span>)</span> iff ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:22_8"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fm2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fb</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">fm2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_8_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:22_8_4"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fm2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fb</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="convex2.html#E7:22"><i><font color="Green">E41</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_8_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_8_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_8_4_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fm2</font> or  ex <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fb</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fm2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">m</font> ) )
 </span><b>by </b><i><a class="txt" href="convex2.html#E7:22"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_8_4_1_1_1"><b>suppose </b></a><a NAME="E1:22_8_4_1_1_1"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fm2</font>
 ;<br/><div class="add"><a NAME="E2:22_8_4_1_1_1"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Fm</font></span>)</span>
 <b>by </b><i><a class="txt" href="convex2.html#E1:22"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:22_8_4_1_1_1"/><b>then </b>
<font color="Maroon">fm2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="convex2.html#E1:22"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E4:22_8_4_1_1_1"/><b>then </b><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">fm2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E5:22_8_4_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fm</font>
 <b>by </b><i><a class="txt" href="convex2.html#E9:22"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E6:22_8_4_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">fm2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="convex2.html#E11:22"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_8_4_1_1_2"><b>suppose </b></a><a NAME="E1:22_8_4_1_1_2"/>
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fb</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fm2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">m</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:22_8_4_1_1_2"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fb</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fm2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> )
 ;<br/><a NAME="E4:22_8_4_1_1_2"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Fb</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><i><font color="Green">E127</font></i>: <span class="p1">(<span class="default"><font color="Maroon">fm2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">fb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i>, <a class="txt" href="convex2.html#E9:22"><i><font color="Green">E43</font></i></a></i>
;<br/><a NAME="E6:22_8_4_1_1_2"/>
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fb</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Fm</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="txt" href="convex2.html#E1:22"><i><font color="Green">E29</font></i></a>, , <a class="txt" href="convex2.html#E9:22"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E7:22_8_4_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">fm2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="convex2.html#E11:22"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:22_8_4"/>
<span class="p1">(<span class="default"><font color="Maroon">fm2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:22_8"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">fm2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span>
 
<b>by </b><i><a class="txt" href="convex2.html#E5:22"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#T20">FUNCT_1:20</a></i>;<br/>
<a NAME="E9:22_8"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rfinseq.html#T16">RFINSEQ:16</a></i>;<br/>
<b>hence </b><a NAME="E10:22_8"/>
<font color="Maroon">f2</font>,<font color="Maroon">fm2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fb</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i>, , , <a class="txt" href="convex2.html#E3:22"><i><font color="Green">E38</font></i></a>, , <a class="txt" href="convex2.html#E7:22"><i><font color="Green">E41</font></i></a>, <a class="ref" href="bagorder.html#T4">BAGORDER:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E54:22"/><b>then </b><i><font color="Green">E129</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">fm2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fb</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfinseq.html#T22">RFINSEQ:22</a></i>;<br/>
<b>set </b><font color="Maroon">F</font> = <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font>;<br/>
<b>set </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span>;<br/>
<a NAME="E55:22"/><i><font color="Green">E130</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font>
 
<b>by </b><i>, , <a class="ref" href="rfinseq.html#T1">RFINSEQ:1</a></i>;<br/>
<a NAME="E56:22"/><i><font color="Green">E131</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fb</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="rlvect_2.html#T65">RLVECT_2:65</a></i>;<br/>
<a NAME="E57:22"/><b>then </b><i><font color="Green">E132</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fb</font>
 
<b>by </b><i><a class="txt" href="convex2.html#E2:22"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E58:22"/>
<font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, , , <a class="ref" href="finseq_3.html#T98">FINSEQ_3:98</a></i>;<br/>
<a NAME="E59:22"/><b>then </b><i><font color="Green">E133</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T98">FINSEQ_3:98</a></i>;<br/>
<a NAME="E60:22"/><i><font color="Green">E134</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font> <a href="tarski.html#R1">c=</a>  <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<i><font color="Green">E135</font></i>:  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="convex2.html#E2:22"><i><font color="Green">E37</font></i></a>, <a class="txt" href="convex2.html#E3:22"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
 <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="convex2.html#E7:22"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>
<a NAME="E62:22"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> &amp;  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_10"><b>proof </b></a><div class="add">

<i><font color="Green">E187</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">fm1</font>
<b>by </b><i><a class="ref" href="euclid_2.html#T8">EUCLID_2:8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span>
<b>by </b><i>, <a class="txt" href="convex2.html#E1:22"><i><font color="Green">E29</font></i></a>, <a class="ref" href="euclid_2.html#T8">EUCLID_2:8</a></i>
;<br/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ft</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_2.html#T8">EUCLID_2:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ft</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fb</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_2.html#T8">EUCLID_2:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ft</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fb</font></span>)</span>
<b>by </b><i><a class="txt" href="convex2.html#E9:22"><i><font color="Green">E43</font></i></a>, <a class="ref" href="integra5.html#T2">INTEGRA5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Ft</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Fm</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Fb</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="euclid_2.html#T8">EUCLID_2:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Fb</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/>
<b>hence </b><a NAME="E3:22_10"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>

 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">ft</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117">RVSUM_1:117</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">ft</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fb</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117">RVSUM_1:117</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">ft</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fb</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="convex2.html#E9:22"><i><font color="Green">E43</font></i></a>, <a class="ref" href="integra5.html#T2">INTEGRA5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">ft</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fm1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fb</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117">RVSUM_1:117</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">ft</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fm1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fb</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117">RVSUM_1:117</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">ft</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fm1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fm2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fb</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fm1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fm2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fb</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span>
<b>by </b><i>, , , , <a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>
<br/>.= 
0 <a href="nat_1.html#K1">+</a> 1

;<br/>
<b>hence </b><a NAME="E5:22_10"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E6:22_10"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_10_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:22_10_4"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_10_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_10_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_10_4_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> or  ex <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">m</font> ) )
 </span><b>by </b><i><a class="txt" href="convex2.html#E11:22"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_10_4_1_1_1"><b>suppose </b></a><a NAME="E1:22_10_4_1_1_1"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:22_10_4_1_1_1"/><b>then </b><i><font color="Green">E190</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_10_4_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_10_4_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_10_4_1_1_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> or  ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font> ) )
 </span><b>by </b><i><a class="txt" href="convex2.html#E12:22"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_10_4_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:22_10_4_1_1_1_1_1_1"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:22_10_4_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ft</font>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T65">RVSUM_1:65</a></i>;<br/><a NAME="E3:22_10_4_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E192</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Ft</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><i><font color="Green">E193</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="txt" href="convex2.html#E13:22"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">ft</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">Ft</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="convex2.html#E16:22"><i><font color="Green">E48</font></i></a></i>
;<br/><a NAME="E5:22_10_4_1_1_1_1_1_1"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Ft</font>
 <b>by </b><i><a class="txt" href="convex2.html#E16:22"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E6:22_10_4_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E195</font></i>: 
<font color="Maroon">Ft</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ft</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E7:22_10_4_1_1_1_1_1_1"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">Ft</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">Lt</font>
 <b>by </b><i>, <a class="txt" href="convex2.html#E22:22"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">Ftn</font> = <font color="Maroon">Ft</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> ;<br/><a NAME="E9:22_10_4_1_1_1_1_1_1"/>
( <font color="Maroon">Ft</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span> &amp; not <font color="Maroon">Ft</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L2</font> )
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E10:22_10_4_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">L2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">Ftn</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>;<br/><a NAME="E11:22_10_4_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">Ftn</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E12:22_10_4_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">Ftn</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="rlvect_2.html#D13">RLVECT_2:def 13</a></i>;<br/><b>then </b><i><font color="Green">E198</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">Ftn</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">Ftn</font></span>)</span>
<b>by </b><i><a class="txt" href="convex2.html#E19:22"><i><font color="Green">E49</font></i></a>, <a class="ref" href="rlvect_2.html#D13">RLVECT_2:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_2.html#D12">RLVECT_2:def 12</a></i>
;<br/> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ft</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_2.html#T8">EUCLID_2:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ft</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="convex2.html#E9:22"><i><font color="Green">E43</font></i></a>, <a class="ref" href="integra5.html#T2">INTEGRA5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Ft</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Fm</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="euclid_2.html#T8">EUCLID_2:8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/><a NAME="E15:22_10_4_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="convex2.html#E12:22"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E16:22_10_4_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E17:22_10_4_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E18:22_10_4_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="convex2.html#E22:22"><i><font color="Green">E50</font></i></a>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E19:22_10_4_1_1_1_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ft</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T42">FINSEQ_1:42</a></i>;<br/><a NAME="E20:22_10_4_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">Ft</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span>
 <b>by </b><i><a class="txt" href="convex2.html#E6"><i><font color="Green">Lemma51</font></i></a></i>;<br/><a NAME="E21:22_10_4_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">Ft</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font>
 <b>by </b><i>, <a class="ref" href="rfinseq.html#T1">RFINSEQ:1</a></i>;<br/><b>then consider </b><font color="Maroon">m'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E23:22_10_4_1_1_1_1_1_1"/><i><font color="Green">E200</font></i>: 
( <font color="Maroon">m'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">m'</font> = <font color="Maroon">m'</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E25:22_10_4_1_1_1_1_1_1"/>
<font color="Maroon">m'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E26:22_10_4_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">m'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E27:22_10_4_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m'</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E28:22_10_4_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="txt" href="convex2.html#E19:22"><i><font color="Green">E49</font></i></a>, , <a class="ref" href="xreal_1.html#T129">XREAL_1:129</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_10_4_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:22_10_4_1_1_1_1_1_2"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:22_10_4_1_1_1_1_1_2"/><i><font color="Green">E201</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> )
 ;<br/> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">fm1</font>
<b>by </b><i><a class="ref" href="euclid_2.html#T8">EUCLID_2:8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span>
<b>by </b><i>, <a class="txt" href="convex2.html#E1:22"><i><font color="Green">E29</font></i></a>, <a class="ref" href="euclid_2.html#T8">EUCLID_2:8</a></i>
;<br/><b>then </b> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span>
<b>by </b><i><a class="ref" href="integra5.html#T2">INTEGRA5:2</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">fm1</font>
<b>by </b><i><a class="ref" href="euclid_2.html#T8">EUCLID_2:8</a></i>
;<br/><a NAME="E6:22_10_4_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E202</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Fm</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="convex2.html#E13:22"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><i><font color="Green">E203</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="txt" href="convex2.html#E13:22"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="txt" href="convex2.html#E13:22"><i><font color="Green">E47</font></i></a>, <a class="ref" href="rvsum_1.html#T26">RVSUM_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">fm1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">fm1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">fm2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">fm2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="convex2.html#E16:22"><i><font color="Green">E48</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="convex2.html#E1:22"><i><font color="Green">E29</font></i></a>, <a class="txt" href="convex2.html#E16:22"><i><font color="Green">E48</font></i></a></i>
;<br/><a NAME="E8:22_10_4_1_1_1_1_1_2"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fm</font>
 <b>by </b><i><a class="txt" href="convex2.html#E16:22"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:22_10_4_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E205</font></i>: 
<font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fm</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">Fmm</font> = <font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/><a NAME="E11:22_10_4_1_1_1_1_1_2"/>
( <font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">Fmm</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> &amp; <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">Fmm</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> )
 <b>by </b><i><a class="ref" href="rlvect_2.html#D13">RLVECT_2:def 13</a></i>;<br/><a NAME="E12:22_10_4_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E207</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="convex2.html#E19:22"><i><font color="Green">E49</font></i></a>, <a class="ref" href="rlvect_2.html#D12">RLVECT_2:def 12</a></i>;<br/> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ft</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_2.html#T8">EUCLID_2:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ft</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="convex2.html#E9:22"><i><font color="Green">E43</font></i></a>, <a class="ref" href="integra5.html#T2">INTEGRA5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Ft</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Fm</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="euclid_2.html#T8">EUCLID_2:8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/><a NAME="E14:22_10_4_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="convex2.html#E12:22"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E15:22_10_4_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E16:22_10_4_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E208</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E17:22_10_4_1_1_1_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Ft</font>
 <b>by </b><i>, <a class="ref" href="euclid_2.html#T8">EUCLID_2:8</a></i>;<br/><b>hence </b><a NAME="E18:22_10_4_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="convex2.html#E13:22"><i><font color="Green">E47</font></i></a>, <a class="txt" href="convex2.html#E22:22"><i><font color="Green">E50</font></i></a>, , , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E19:22_10_4_1_1_1_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fm</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T43">FINSEQ_1:43</a></i>;<br/><a NAME="E20:22_10_4_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span>
 <b>by </b><i><a class="txt" href="convex2.html#E6"><i><font color="Green">Lemma51</font></i></a></i>;<br/><a NAME="E21:22_10_4_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font>
 <b>by </b><i>, <a class="ref" href="rfinseq.html#T1">RFINSEQ:1</a></i>;<br/><b>then consider </b><font color="Maroon">m'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E23:22_10_4_1_1_1_1_1_2"/><i><font color="Green">E209</font></i>: 
( <font color="Maroon">m'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">m'</font> = <font color="Maroon">m'</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E25:22_10_4_1_1_1_1_1_2"/>
<font color="Maroon">m'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E26:22_10_4_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">m'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E27:22_10_4_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m'</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i/>;<br/><a NAME="E28:22_10_4_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E210</font></i>: 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T129">XREAL_1:129</a></i>;<br/><a NAME="E29:22_10_4_1_1_1_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fm</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T42">FINSEQ_1:42</a></i>;<br/><a NAME="E30:22_10_4_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span>
 <b>by </b><i><a class="txt" href="convex2.html#E6"><i><font color="Green">Lemma51</font></i></a></i>;<br/><a NAME="E31:22_10_4_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F2</font>
 <b>by </b><i>, <a class="ref" href="rfinseq.html#T1">RFINSEQ:1</a></i>;<br/><b>then consider </b><font color="Maroon">m'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E33:22_10_4_1_1_1_1_1_2"/><i><font color="Green">E211</font></i>: 
( <font color="Maroon">m'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">m'</font> = <font color="Maroon">m'</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E35:22_10_4_1_1_1_1_1_2"/>
<font color="Maroon">m'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E36:22_10_4_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">m'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E37:22_10_4_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m'</font></span>)</span> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i/>;<br/><a NAME="E38:22_10_4_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T129">XREAL_1:129</a></i>;<br/><a NAME="E39:22_10_4_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">Fm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E40:22_10_4_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="convex2.html#E19:22"><i><font color="Green">E49</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:22_10_4_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_10_4_1_1_2"><b>suppose </b></a><a NAME="E1:22_10_4_1_1_2"/>
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">m</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:22_10_4_1_1_2"/><i><font color="Green">E212</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> )
 ;<br/><a NAME="E4:22_10_4_1_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fb</font>
 <b>by </b><i><a class="txt" href="convex2.html#E12:22"><i><font color="Green">E45</font></i></a>, <a class="ref" href="rvsum_1.html#T65">RVSUM_1:65</a></i>;<br/><a NAME="E5:22_10_4_1_1_2"/><b>then </b><i><font color="Green">E213</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Fb</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><i><font color="Green">E214</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="txt" href="convex2.html#E12:22"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">fb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">Fb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/><a NAME="E7:22_10_4_1_1_2"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fb</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E8:22_10_4_1_1_2"/><b>then </b><i><font color="Green">E216</font></i>: 
<font color="Maroon">Fb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fb</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">Fbm</font> = <font color="Maroon">Fb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/><a NAME="E10:22_10_4_1_1_2"/>
( <font color="Maroon">Fb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span> &amp; not <font color="Maroon">Fb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font> )
 <b>by </b><i>, , , <a class="txt" href="convex2.html#E19:22"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E11:22_10_4_1_1_2"/><b>then </b>
<font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">Fbm</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>;<br/><a NAME="E12:22_10_4_1_1_2"/><b>then </b>
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">Fbm</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E13:22_10_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">Fbm</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="rlvect_2.html#D13">RLVECT_2:def 13</a></i>;<br/><b>then </b><i><font color="Green">E218</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">Fbm</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">Fbm</font></span>)</span>
<b>by </b><i><a class="txt" href="convex2.html#E13:22"><i><font color="Green">E47</font></i></a>, <a class="ref" href="rlvect_2.html#D13">RLVECT_2:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_2.html#D12">RLVECT_2:def 12</a></i>
;<br/> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ft</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_2.html#T8">EUCLID_2:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ft</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="convex2.html#E9:22"><i><font color="Green">E43</font></i></a>, <a class="ref" href="integra5.html#T2">INTEGRA5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Ft</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Fm</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="euclid_2.html#T8">EUCLID_2:8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/><b>hence </b><a NAME="E16:22_10_4_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="convex2.html#E12:22"><i><font color="Green">E45</font></i></a>, <a class="txt" href="convex2.html#E16:22"><i><font color="Green">E48</font></i></a>, <a class="txt" href="convex2.html#E22:22"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E17:22_10_4_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fb</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T43">FINSEQ_1:43</a></i>;<br/><a NAME="E18:22_10_4_1_1_2"/><b>then </b>
<font color="Maroon">Fb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Fm</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span>
 <b>by </b><i><a class="txt" href="convex2.html#E19:22"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E19:22_10_4_1_1_2"/><b>then </b>
<font color="Maroon">Fb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F2</font>
 <b>by </b><i>, <a class="ref" href="rfinseq.html#T1">RFINSEQ:1</a></i>;<br/><b>then consider </b><font color="Maroon">m'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E21:22_10_4_1_1_2"/><i><font color="Green">E219</font></i>: 
( <font color="Maroon">m'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">m'</font> = <font color="Maroon">m'</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="convex2.html#E6"><i><font color="Green">Lemma51</font></i></a></i>;<br/><a NAME="E23:22_10_4_1_1_2"/>
<font color="Maroon">m'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font></span>)</span>
 <b>by </b><i><a class="txt" href="convex2.html#E6"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E24:22_10_4_1_1_2"/><b>then </b>
<font color="Maroon">m'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E25:22_10_4_1_1_2"/><b>then </b>
( <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m'</font></span>)</span> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E26:22_10_4_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="txt" href="convex2.html#E13:22"><i><font color="Green">E47</font></i></a>, <a class="txt" href="convex2.html#E6"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="xreal_1.html#T129">XREAL_1:129</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:22_10_4"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:22_10"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Ft</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fm</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">ft</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fm2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">fb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E63:22"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">L2</font></span>)</span> is   <a href="convex2.html#NM1">Convex_Combination</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="convex2.html#E5:22"><i><font color="Green">E40</font></i></a>, , <a class="ref" href="convex1.html#D3">CONVEX1:def 3</a></i>;<br/>


</div>
