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

<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">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/>the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Olive" title="b2">g</font>;<br/>
<a NAME="E1:37"/><i><font color="Green" title="E32">A1</font></i>: 
<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="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font>;<br/>

<b>assume </b><a NAME="E1:37_1"/>
( <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a>  &amp;  <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="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>

<a NAME="E2:37_1"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">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="c2">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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a>, <a class="ref" href="rfinseq.html#T16" title="RFINSEQ:th.16">RFINSEQ:16</a></i>;<br/>
<b>hence </b><a NAME="E3:37_1"/>
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Maroon" title="c3">g</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:37"/><i><font color="Green" title="E33">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">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#K1" title="NAT_1:func.1">+</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" title="c2">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:37_2"/><i><font color="Green" title="E34">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font>;<br/>

<b>assume </b><a NAME="E2:37_2"/><i><font color="Green" title="E35">A4</font></i>: 
( <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/>

<a NAME="E3:37_2"/><b>then </b><i><font color="Green" title="E36">A5</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">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="c4">g</font> )
 
<b>by </b><i><a class="ref" href="rfinseq.html#T1" title="RFINSEQ:th.1">RFINSEQ:1</a>, <a class="ref" href="rfinseq.html#T16" title="RFINSEQ:th.16">RFINSEQ:16</a></i>;<br/>
<a NAME="E4:37_2"/>
<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="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:37_2"/><b>then </b><i><font color="Green" title="E37">A6</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c3">f</font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E2:37_2"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:37_2"/><b>then </b><i><font color="Green" title="E38">A7</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:37_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> <b>by </b><i><a class="txt" href="matrix_7.html#E6:37_2"><i><font color="Green" title="E38">A7</font></i></a></i>;<br/>
<a NAME="E9:37_2"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E3:37_2"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="matrix_7.html#E5:37_2"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E11:37_2"/><i><font color="Green" title="E39">A8</font></i>: 
( <font color="Maroon" title="c6">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="c4">g</font> &amp; <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E12:37_2"/><i><font color="Green" title="E40">A9</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">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="c4">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c6">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfinseq.html#T21" title="RFINSEQ:th.21">RFINSEQ:21</a></i>;<br/>
<a NAME="E13:37_2"/><i><font color="Green" title="E41">A10</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> &amp; <font color="Maroon" title="c6">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="c4">g</font> )
 
<b>by </b><i><a class="txt" href="matrix_7.html#E11:37_2"><i><font color="Green" title="E39">A8</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E14:37_2"/><b>then </b>
 <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="c6">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T4" title="FINSEQ_2:th.4">FINSEQ_2:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">m1</font> = <font color="Maroon" title="c6">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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>  <b>by </b><i><a class="ref" href="finseq_2.html#T5" title="FINSEQ_2:th.5">FINSEQ_2:5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">gg</font> = <font color="Maroon" title="c4">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c6">m</font>;<br/>
<b>set </b><font color="Maroon" title="c9">gm</font> = <font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">m</font>;<br/>
<a NAME="E16:37_2"/><i><font color="Green" title="E42">A11</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E13:37_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E17:37_2"/><i><font color="Green" title="E43">A12</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E18:37_2"/>
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E19:37_2"/><b>then </b>
<font color="Maroon" title="c6">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="c6">m</font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E13:37_2"><i><font color="Green" title="E41">A10</font></i></a></i>;<br/>
<a NAME="E20:37_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c6">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="c4">g</font> )
 
<b>by </b><i><a class="txt" href="matrix_7.html#E11:37_2"><i><font color="Green" title="E39">A8</font></i></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></i>;<br/>
<a NAME="E21:37_2"/><b>then </b><i><font color="Green" title="E44">A13</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">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="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">m1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E16:37_2"><i><font color="Green" title="E42">A11</font></i></a>, <a class="txt" href="matrix_7.html#E17:37_2"><i><font color="Green" title="E43">A12</font></i></a>, <a class="ref" href="rfinseq.html#T20" title="RFINSEQ:th.20">RFINSEQ:20</a></i>;<br/>
<a NAME="E22:37_2"/>
<font color="Maroon" title="c7">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E17:37_2"><i><font color="Green" title="E43">A12</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E23:37_2"/><b>then </b><i><font color="Green" title="E45">A14</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">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="c6">m</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/>
<i><font color="Green" title="E46">A15</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">m1</font> = 
<font color="Maroon" title="c4">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="c6">m</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">m1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T100" title="RELAT_1:th.100">RELAT_1:100</a></i>
<br/>.= 
<font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">m1</font>
<b>by </b><i><a class="txt" href="matrix_7.html#E23:37_2"><i><font color="Green" title="E45">A14</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c10">n1</font> = <font color="Maroon" title="c2">n</font> 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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">fn</font> = <font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">n1</font>;<br/>
<a NAME="E26:37_2"/>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E27:37_2"/><b>then </b><i><font color="Green" title="E47">A16</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E2:37_2"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E28:37_2"/><i><font color="Green" title="E48">A17</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">n1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E2:37_2"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="rfinseq.html#T20" title="RFINSEQ:th.20">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" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><a NAME="E1:37_2_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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"><font color="Maroon" title="c4">g</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E2:37_2"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="rfinseq.html#D1" title="RFINSEQ:def.1">RFINSEQ:def 1</a></i>;<br/><b>then </b><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="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">n1</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">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="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">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="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">m1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&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="c4">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="matrix_7.html#E12:37_2"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="matrix_7.html#E21:37_2"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="matrix_7.html#E24:37_2"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="matrix_7.html#E28:37_2"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="finseq_3.html#T63" title="FINSEQ_3:th.63">FINSEQ_3:63</a></i>
<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="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">m1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">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="c4">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_3.html#T63" title="FINSEQ_3:th.63">FINSEQ_3:63</a></i>
<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="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">m1</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">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="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">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="c4">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_3.html#T63" title="FINSEQ_3:th.63">FINSEQ_3:63</a></i>
<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="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">m1</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">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="c4">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">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="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">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="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">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="c4">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">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="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_3.html#T63" title="FINSEQ_3:th.63">FINSEQ_3:63</a></i>
;<br/><b>hence </b><a NAME="E3:37_2_2"/>
 <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"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">n1</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">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="c4">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 ;<br/></div><b>end;</b></div>
<a NAME="E30:37_2"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">n1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">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="c4">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="ref" href="rfinseq.html#D1" title="RFINSEQ:def.1">RFINSEQ:def 1</a></i>;<br/>
<a NAME="E31:37_2"/><b>then </b>
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">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="c4">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E1:37_2"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="matrix_7.html#E27:37_2"><i><font color="Green" title="E47">A16</font></i></a></i>;<br/>
<b>hence </b>the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Maroon" title="c3">f</font> = 
<span class="p1">(<span class="default">the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">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="c4">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="matrix_7.html#E28:37_2"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="matrix_7.html#T30" target="_self" title="MATRIX_7:th.30">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">m1</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default">the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_7.html#T30" target="_self" title="MATRIX_7:th.30">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">m1</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default">the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix_7.html#E21:37_2"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="matrix_7.html#E24:37_2"><i><font color="Green" title="E46">A15</font></i></a>, <a class="ref" href="matrix_7.html#T30" target="_self" title="MATRIX_7:th.30">Th30</a></i>
<br/>.= 
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Maroon" title="c4">g</font>
<b>by </b><i><a class="txt" href="matrix_7.html#E12:37_2"><i><font color="Green" title="E40">A9</font></i></a>, <a class="ref" href="matrix_7.html#T30" target="_self" title="MATRIX_7:th.30">Th30</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E3:37"/><i><font color="Green" title="E34">A18</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="matrix_7.html#E1:37"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="matrix_7.html#E2:37"><i><font color="Green" title="E33">A2</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c2">R1</font>, <font color="Maroon" title="c3">R2</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font>;<br/>

<b>assume </b><a NAME="E4:37"/><i><font color="Green" title="E35">A19</font></i>: 
<font color="Maroon" title="c2">R1</font>,<font color="Maroon" title="c3">R2</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a> 
 ;<br/>

<a NAME="E5:37"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">R1</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">R1</font>
 
;<br/>
<b>hence </b><a NAME="E6:37"/>
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Maroon" title="c2">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Maroon" title="c3">R2</font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E3:37"><i><font color="Green" title="E34">A18</font></i></a>, <a class="txt" href="matrix_7.html#E4:37"><i><font color="Green" title="E35">A19</font></i></a></i>;<br/>


</div>
