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

<span class="kw">let </span><font color="Maroon" title="c1">D</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> <font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  holds <br/> <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Olive" title="b1">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Olive" title="b2">g</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> <font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Olive" title="b1">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Olive" title="b2">g</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> <font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  implies  <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c3">g</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:43"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c3">g</font></span><br/>

<a NAME="E2:43"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font>
 
;<br/>
<a NAME="E3:43"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_1"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> <font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies  <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c6">g</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:43_1"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 <span class="kw">and </span><br/><a NAME="E3:43_1"/><span class="lab"><font color="Green" title="E30">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c6">g</font></span><br/>

<a NAME="E4:43_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E5:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A7</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> <font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></span>;<br/>
<a NAME="E7:43_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="classes1.html#T83" title="CLASSES1:th.83">CLASSES1:83</a></span>;<br/>
<a NAME="E8:43_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E10:43_1"/><span class="lab"><font color="Green" title="E32">A8</font></span>: 
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g</font>
 <span class="kw">and </span><br/><a NAME="E11:43_1"/><span class="lab"><font color="Green" title="E33">A9</font></span>: 
<font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></span>;<br/>
<a NAME="E12:43_1"/><span class="lab"><font color="Green" title="E34">A10</font></span>: 
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c8">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#T21" title="RFINSEQ:th.21">RFINSEQ:21</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">gg</font> = <font color="Maroon" title="c6">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c8">m</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">gm</font> = <font color="Maroon" title="c6">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font>;<br/>
<a NAME="E13:43_1"/>
<font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E14:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A11</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">fn</font> = <font color="Maroon" title="c5">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">n</font>;<br/>
<a NAME="E15:43_1"/><span class="lab"><font color="Green" title="E36">A12</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="rfinseq.html#T20" title="RFINSEQ:th.20">RFINSEQ:20</a></span>;<br/>
<a NAME="E16:43_1"/><span class="lab"><font color="Green" title="E37">A13</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E17:43_1"/><span class="kw">then </span>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T4" title="FINSEQ_2:th.4">FINSEQ_2:4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">m1</font> = <font color="Maroon" title="c8">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T5" title="FINSEQ_2:th.5">FINSEQ_2:5</a></span>;<br/>
<a NAME="E19:43_1"/><span class="lab"><font color="Green" title="E38">A14</font></span>: 
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E20:43_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E21:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A15</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c12">m1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></span>;<br/>
<a NAME="E22:43_1"/>
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:43_1"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E23:43_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E11:43_1"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></span>;<br/>
<a NAME="E24:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A16</font></span>: 
<font color="Maroon" title="c6">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">m1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:43_1"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E19:43_1"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="ref" href="rfinseq.html#T20" title="RFINSEQ:th.20">RFINSEQ:20</a></span>;<br/>
<span class="lab"><font color="Green" title="E41">A17</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">m1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c12">m1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c12">m1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>
<br/>.= 
<font color="Maroon" title="c6">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T100" title="RELAT_1:th.100">RELAT_1:100</a></span>
<br/>.= 
<font color="Maroon" title="c6">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c12">m1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:43_1"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
<br/>.= 
<font color="Maroon" title="c6">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">m1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">n</font></span>)</span>,<font color="Maroon" title="c13">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">m1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">x</font></span>)</span></span><br/><a NAME="E1:43_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c13">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c13">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="classes1.html#D9" title="CLASSES1:def.9">CLASSES1:def 9</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">m1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E24:43_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E25:43_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E15:43_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="ref" href="finseq_3.html#T63" title="FINSEQ_3:th.63">FINSEQ_3:63</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">m1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T63" title="FINSEQ_3:th.63">FINSEQ_3:63</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">m1</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T63" title="FINSEQ_3:th.63">FINSEQ_3:63</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">m1</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">m1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T63" title="FINSEQ_3:th.63">FINSEQ_3:63</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:43_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">n</font></span>)</span>,<font color="Maroon" title="c13">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">m1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">x</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E27:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A18</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">m1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="classes1.html#D9" title="CLASSES1:def.9">CLASSES1:def 9</a></span>;<br/>
<a NAME="E28:43_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E29:43_1"/><span class="kw">then </span>
 <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">m1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E27:43_1"><span class="lab"><font color="Green" title="E42">A18</font></span></a></span>;<br/>
<span class="kw">hence </span> <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c5">f</font> = 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">m1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="rfunct_3.html#K6" title="RFUNCT_3:func.6">+</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:43_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="ref" href="rfunct_3.html#T24" target="_self" title="RFUNCT_3:th.24">Th24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span> <a href="rfunct_3.html#K6" title="RFUNCT_3:func.6">+</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="rfunct_3.html#K6" title="RFUNCT_3:func.6">+</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_3.html#T24" target="_self" title="RFUNCT_3:th.24">Th24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span> <a href="rfunct_3.html#K6" title="RFUNCT_3:func.6">+</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="rfunct_3.html#K6" title="RFUNCT_3:func.6">+</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T19" title="RFUNCT_1:th.19">RFUNCT_1:19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="rfunct_3.html#K6" title="RFUNCT_3:func.6">+</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:43_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E25:43_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="ref" href="rfunct_3.html#T24" target="_self" title="RFUNCT_3:th.24">Th24</a></span>
<br/>.= 
 <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c6">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="ref" href="rfunct_3.html#T24" target="_self" title="RFUNCT_3:th.24">Th24</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:43"/><span class="lab"><font color="Green" title="E28">A19</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> <font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c5">g</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_2"/>
( <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c5">g</font></span><br/>

<a NAME="E2:43_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">X</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> <font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#T16" title="RFINSEQ:th.16">RFINSEQ:16</a></span>;<br/>
<a NAME="E3:43_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> <font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span>
 
;<br/>
<span class="kw">hence </span><a NAME="E4:43_2"/>
 <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c5">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_2"><span class="lab"><font color="Green" title="E29">X</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:43"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E28">A19</font></span></a>, <a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E27">A3</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:43"/>
 <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c3">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E26">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
