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<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being   <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st $1 <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="Olive" title="b1">g1</font> &amp; <font color="Olive" title="b1">g1</font>,<font color="Olive" title="b2">g2</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  holds <br/><font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g2</font>;<br/>
<a NAME="E1:35"/><span class="lab"><font color="Green" title="E26">A1</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>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c1">n</font>] implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c1">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">g1</font>, <font color="Maroon" title="c3">g2</font> be   <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <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">g1</font> &amp; <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c3">g2</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  implies <font color="Maroon" title="c2">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g2</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">r</font> = <font color="Maroon" title="c2">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">g1n</font> = <font color="Maroon" title="c2">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c6">g2n</font> = <font color="Maroon" title="c3">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">n</font> as   <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#T20" target="_self" title="RFINSEQ:th.20">Th33</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:35_1"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 <span class="kw">and </span><br/><a NAME="E4:35_1"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c3">g2</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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g2</font></span><br/>

<a NAME="E5:35_1"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g2</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">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="rfinseq.html#T3" target="_self" title="RFINSEQ:th.3">Th16</a></span>;<br/>
<a NAME="E6:35_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A6</font></span>: 
<font color="Maroon" title="c2">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E4:35_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="rfinseq.html#E28"><span class="lab"><font color="Green" title="E21">Lm4</font></span></a></span>;<br/>
<a NAME="E7:35_1"/><span class="lab"><font color="Green" title="E32">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="rfinseq.html#T7" target="_self" title="RFINSEQ:th.7">Th20</a></span>;<br/>
<a NAME="E8:35_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E9:35_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">g1n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g2n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E3:35_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E4:35_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E5:35_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="rfinseq.html#E28"><span class="lab"><font color="Green" title="E21">Lm4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:35_1"/>
<font color="Maroon" title="c2">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E5:35_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E6:35_1"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="txt" href="#E7:35_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="ref" href="rfinseq.html#T7" target="_self" title="RFINSEQ:th.7">Th20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:35"/><span class="lab"><font color="Green" title="E27">A8</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">g1</font>, <font color="Maroon" title="c2">g2</font> be   <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c1">g1</font> &amp; <font color="Maroon" title="c1">g1</font>,<font color="Maroon" title="c2">g2</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  implies <font color="Maroon" title="c1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">g1</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="c1">g1</font>,<font color="Maroon" title="c2">g2</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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g2</font></span><br/>

<a NAME="E2:35_2"/><span class="kw">then </span>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g2</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="c1">g1</font> &amp; <font color="Maroon" title="c1">g1</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> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#T3" target="_self" title="RFINSEQ:th.3">Th16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:35_2"/>
<font color="Maroon" title="c1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E3:35"/>
 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>3</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="#E2:35"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E26">A1</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
