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<b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">g</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:19"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">P</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font><b> such that </b><br/><a NAME="E3:19"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="rfinseq.html#T17">RFINSEQ:17</a></i>;<br/>
<a NAME="E4:19"/><i><font color="Green">E30</font></i>: 
(  <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> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, <a class="ref" href="rfinseq.html#T16">RFINSEQ:16</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1"><b>suppose </b></a><a NAME="E1:19_1_1"/><i><font color="Green">E31</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:19_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="rfinseq2.html#K2" target="_self">min_p</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:19_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E4:19_1_1"/><i><font color="Green">E36</font></i>: 
 <a href="rfinseq2.html#K2" target="_self">min_p</a> <font color="Maroon">f</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="ref" href="rfinseq.html#T16">RFINSEQ:16</a></i>;<br/><a NAME="E5:19_1_1"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E6:19_1_1"/><i><font color="Green">E38</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E7:19_1_1"/><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/><a NAME="E8:19_1_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:19_1_1"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E10:19_1_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E11:19_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <font color="Maroon">f</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E13:19_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</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="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E14:19_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="rfinseq2.html#K4" target="_self">min</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E15:19_1_1"/>
 <a href="rfinseq2.html#K4" target="_self">min</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="rfinseq2.html#K4" target="_self">min</a> <font color="Maroon">g</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2"><b>suppose </b></a><a NAME="E1:19_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:19_1_2"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E3:19_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><b>hence </b><a NAME="E4:19_1_2"/>
 <a href="rfinseq2.html#K4" target="_self">min</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="rfinseq2.html#K4" target="_self">min</a> <font color="Maroon">g</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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