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

<b>let </b><font color="Maroon">K</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font>  st <font color="Olive">f</font>,<font color="Olive">g</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <font color="Olive">g</font>;<br/>
<a NAME="E1:37"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font>, <font color="Maroon">g</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font>;<br/>

<b>assume </b><a NAME="E1:37_1"/>
( <font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<a NAME="E2:37_1"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a>, <a class="ref" href="rfinseq.html#T16">RFINSEQ:16</a></i>;<br/>
<b>hence </b><a NAME="E3:37_1"/>
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:37"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:37_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font>, <font color="Maroon">g</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font>;<br/>

<b>assume </b><a NAME="E2:37_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 ;<br/>

<a NAME="E3:37_2"/><b>then </b><i><font color="Green">E44</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 
<b>by </b><i><a class="ref" href="rfinseq.html#T1">RFINSEQ:1</a>, <a class="ref" href="rfinseq.html#T16">RFINSEQ:16</a></i>;<br/>
<a NAME="E4:37_2"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:37_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:37_2"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:37_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> <b>by </b><i/>;<br/>
<a NAME="E9:37_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E1:37"><i><font color="Green">E39</font></i></a>, <a class="txt" href="matrix_7.html#E2:37"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<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="E11:37_2"/><i><font color="Green">E47</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">g</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E12:37_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfinseq.html#T21">RFINSEQ:21</a></i>;<br/>
<a NAME="E13:37_2"/><i><font color="Green">E53</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 
<b>by </b><i><a class="txt" href="matrix_7.html#E1:37_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E14:37_2"/><b>then </b>
 <a href="xxreal_0.html#K4">max</a> 0,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T4">FINSEQ_2:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T5">FINSEQ_2:5</a></i>;<br/>
<b>set </b><font color="Maroon">gg</font> = <font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">m</font>;<br/>
<b>set </b><font color="Maroon">gm</font> = <font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font>;<br/>
<a NAME="E16:37_2"/><i><font color="Green">E54</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E3:37_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E17:37_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E18:37_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E3:37_2"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E19:37_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> )
 
<b>by </b><i><a class="txt" href="matrix_7.html#E1:37_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a></i>;<br/>
<a NAME="E20:37_2"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E5:37_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="matrix_7.html#E6:37_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="rfinseq.html#T20">RFINSEQ:20</a></i>;<br/>
<a NAME="E21:37_2"/>
<font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E6:37_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E22:37_2"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<i><font color="Green">E59</font></i>: <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m1</font> = 
<font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T100">RELAT_1:100</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m1</font>
<b>by </b><i><a class="ref" href="matrix_7.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>set </b><font color="Maroon">fn</font> = <font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font>;<br/>
<a NAME="E25:37_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E26:37_2"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E27:37_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i>, <a class="ref" href="rfinseq.html#T20">RFINSEQ:20</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:37_2_2"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="rfinseq.html#D1">RFINSEQ:def 1</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="binarith.html#K11">*&gt;</a></span> <a href="relat_1.html#K10">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> = 
 <a href="card_1.html#K4">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">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="matrix_7.html#E2:37_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="matrix_7.html#E11:37_2"><i><font color="Green">E47</font></i></a>, , , <a class="ref" href="finseq_3.html#T63">FINSEQ_3:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">m</font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_3.html#T63">FINSEQ_3:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m1</font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="binarith.html#K11">*&gt;</a></span> <a href="relat_1.html#K10">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">m</font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_3.html#T63">FINSEQ_3:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m1</font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">m</font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="binarith.html#K11">*&gt;</a></span> <a href="relat_1.html#K10">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="binarith.html#K11">*&gt;</a></span> <a href="relat_1.html#K10">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_3.html#T63">FINSEQ_3:63</a></i>
;<br/><b>hence </b><a NAME="E3:37_2_2"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 ;<br/></div><b>end;</b></div>
<a NAME="E29:37_2"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">m</font></span>)</span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="ref" href="rfinseq.html#D1">RFINSEQ:def 1</a></i>;<br/>
<a NAME="E30:37_2"/><b>then </b>
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_7.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>hence </b>the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix_7.html#E11:37_2"><i><font color="Green">E47</font></i></a>, , </i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <font color="Maroon">g</font>
<b>by </b><i><a class="txt" href="matrix_7.html#E2:37_2"><i><font color="Green">E43</font></i></a>, </i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E3:37"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>let </b><font color="Maroon">R1</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font>, <font color="Maroon">R2</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font>;<br/>

<b>assume </b><a NAME="E4:37"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">R1</font>,<font color="Maroon">R2</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 ;<br/>

<a NAME="E5:37"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">R1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R1</font>
 
;<br/>
<b>hence </b><a NAME="E6:37"/>
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <font color="Maroon">R1</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <font color="Maroon">R2</font>
 <b>by </b><i><a class="ref" href="matrix_7.html#T1" target="_self">Th1</a>, </i>;<br/>


</div>
