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<b>let </b><font color="Maroon">p1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="rfunct_3.html#K4" target="_self">PFuncs</a> <font color="Maroon">D</font>,<a href="numbers.html#K1">REAL</a> , <font color="Maroon">p2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="rfunct_3.html#K4" target="_self">PFuncs</a> <font color="Maroon">D</font>,<a href="numbers.html#K1">REAL</a> ;<br/>

<b>set </b><font color="Maroon">m</font> =  <a href="rfunct_3.html#K1" target="_self">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:41_1_2"/><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="rfunct_3.html#K1" target="_self">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>and </b><br/><a NAME="E2:41_1_2"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">p1</font> holds <br/>for <font color="Olive">F</font> being  <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">D</font>, <a href="numbers.html#K1">REAL</a> <br/> for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> holds <br/><font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Olive">F</font>
 <b>and </b><br/><a NAME="E3:41_1_2"/><i><font color="Green">E36</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a>  <a href="rfunct_3.html#K1" target="_self">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>and </b><br/><a NAME="E4:41_1_2"/><i><font color="Green">E94</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">p2</font> holds <br/>for <font color="Olive">F</font> being  <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">D</font>, <a href="numbers.html#K1">REAL</a> <br/> for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> holds <br/><font color="Maroon">p2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Olive">F</font>
 ;<br/>

<a NAME="E5:41_1_2"/><i><font color="Green">E95</font></i>: 
(  <a href="rfunct_3.html#K1" target="_self">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> &amp;  <a href="rfunct_3.html#K1" target="_self">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <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="square_1.html#T35">SQUARE_1:35</a></i>;<br/>
<a NAME="E6:41_1_2"/><i><font color="Green">E99</font></i>: 
(  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font></span>)</span> &amp;  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:41_1_2_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K1" target="_self">min</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:41_1_2_1"/><b>then </b><i><font color="Green">E101</font></i>: 
( 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="rfunct_3.html#K1" target="_self">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:41_1_2_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</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">f</font> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E4:41_1_2_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">R</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="txt" href="rfunct_3.html#E1:41_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as    <a href="rfunct_3.html#M2" target="_self">Element</a> of  <a href="rfunct_3.html#K4" target="_self">PFuncs</a> <font color="Maroon">D</font>,<a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E7:41_1_2_1"/>
( <font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="rfunct_3.html#K13" target="_self">(#)</a> <font color="Maroon">F</font> &amp; <font color="Maroon">p2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="rfunct_3.html#K13" target="_self">(#)</a> <font color="Maroon">F</font> )
 <b>by </b><i>, , <a class="ref" href="rfunct_3.html#T1" target="_self">Th1</a>, , , </i>;<br/><b>hence </b><a NAME="E8:41_1_2_1"/>
<font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:41_1_2"/>
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 <b>by </b><i>, <a class="ref" href="rfunct_3.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>


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