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<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">C</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">D</font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">B</font> be   <a href="rearran1.html#NM1">RearrangmentGen</a> of <font color="Maroon">C</font>;<br/>









<b>assume </b><a NAME="E1:50"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">F</font> is <a href="partfun1.html#V1">total</a> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">C</font> )
 ;<br/>

<b>set </b><font color="Maroon">b</font> =  <a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font>;<br/>
<a NAME="E2:50"/><i><font color="Green">E34</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="rearran1.html#D1" target="_self">Def1</a>, <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<a NAME="E3:50"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D12">RFUNCT_3:def 12</a></i>;<br/>
<a NAME="E4:50"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">F</font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">r</font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<a NAME="E5:50"/><b>then </b><i><font color="Green">E44</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="ref" href="rearran1.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E6:50"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D12">RFUNCT_3:def 12</a></i>;<br/>
<a NAME="E7:50"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">F</font> <a href="rfunct_3.html#K23">|</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">r</font></span>)</span> <a href="rfunct_3.html#K23">|</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">r</font> )
 
<b>by </b><i>, <a class="ref" href="rearran1.html#D2" target="_self">Def2</a>, <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/>
<a NAME="E8:50"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="rfunct_3.html#K22">FinS</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">D</font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rfunct_3.html#T76">RFUNCT_3:76</a></i>;<br/>
<a NAME="E9:50"/><i><font color="Green">E52</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">D</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">D</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">D</font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">D</font> )
 
<b>by </b><i>, <a class="ref" href="rearran1.html#D2" target="_self">Def2</a>, <a class="txt" href="rearran1.html#E3:50"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a>, <a class="ref" href="rfunct_3.html#T70">RFUNCT_3:70</a></i>;<br/>
<a NAME="E10:50"/><i><font color="Green">E54</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">C</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">C</font> )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>;<br/><b>assume </b><a NAME="E1:50_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">B</font></span>)</span>
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/><a NAME="E2:50_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/><a NAME="E3:50_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span>
 ;<br/><a NAME="E4:50_1"/><b>then </b><i><font color="Green">E55</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:50_1"/><b>then </b><i><font color="Green">E56</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:50_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E7:50_1"/><b>then </b><i><font color="Green">E57</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">mi</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:50_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">mi</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i><a NAME="E10:50_1"/><i><font color="Green">E59</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi</font> &amp; <font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> )
 <b>by </b><i><a class="txt" href="rearran1.html#E7:50"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:50_1"/><b>then </b>
 <a href="xxreal_0.html#K4">max</a> 0,<span class="p1">(<span class="default"><font color="Maroon">mi</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">mi</font> <a href="real_1.html#K5">-</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">mi</font> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">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/><a NAME="E13:50_1"/>
( <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 &amp; <font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">mi</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E14:50_1"/><b>then </b>
( <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi</font> &amp; <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">mi</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E15:50_1"/><b>then </b><i><font color="Green">E60</font></i>: 
( <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> &amp; <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1_1_1"/>
( <font color="Maroon">mi</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">mi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1"><b>case </b></a><a NAME="E1:50_1_1_1_1"/>
<font color="Maroon">mi</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:50_1_1_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
( <span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">B</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> 1 &amp; <span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> 1 )
 <b>by </b><i><a class="ref" href="rearran1.html#D1" target="_self">Def1</a>, <a class="txt" href="rearran1.html#E4"><i><font color="Green">Lemma39</font></i></a>, <a class="txt" href="rearran1.html#E7:50"><i><font color="Green">E49</font></i></a>, </i>;<br/><a NAME="E3:50_1_1_1_1"/><i><font color="Green">E63</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="rearran1.html#D1" target="_self">Def1</a>, <a class="txt" href="rearran1.html#E5:50"><i><font color="Green">E44</font></i></a>, <a class="txt" href="rearran1.html#E6:50"><i><font color="Green">E45</font></i></a>, <a class="txt" href="rearran1.html#E6"><i><font color="Green">Lemma46</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:50_1_1_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="rearran1.html#D1" target="_self">Def1</a>, <a class="txt" href="rearran1.html#E6:50"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:50_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">D</font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">B</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font>
<b>by </b><i><a class="txt" href="rearran1.html#E4:50"><i><font color="Green">E43</font></i></a>, <a class="txt" href="rearran1.html#E9:50"><i><font color="Green">E52</font></i></a>, , <a class="ref" href="rvsum_1.html#T47">RVSUM_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font>
<b>by </b><i><a class="txt" href="rearran1.html#E5"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="rfunct_3.html#D12">RFUNCT_3:def 12</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_2"><b>case </b></a><a NAME="E1:50_1_1_1_2"/>
<font color="Maroon">mi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:50_1_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">mi</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:50_1_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:50_1_1_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E5:50_1_1_1_2"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E8:50"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_1_1_2_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font>
 ;<br/><a NAME="E2:50_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font> &amp; <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">mi</font> )
 <b>by </b><i><a class="txt" href="rearran1.html#E7:50"><i><font color="Green">E49</font></i></a>, , <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/><b>hence </b><a NAME="E3:50_1_1_1_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E7:50_1_1_1_2"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E7:50"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E8:50_1_1_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
( <span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">B</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="rearran1.html#D1" target="_self">Def1</a>, <a class="txt" href="rearran1.html#E4"><i><font color="Green">Lemma39</font></i></a>, <a class="txt" href="rearran1.html#E8:50"><i><font color="Green">E51</font></i></a>, <a class="txt" href="rearran1.html#E9:50"><i><font color="Green">E52</font></i></a>, </i>;<br/><a NAME="E9:50_1_1_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="rearran1.html#D1" target="_self">Def1</a>, <a class="txt" href="rearran1.html#E5:50"><i><font color="Green">E44</font></i></a>, <a class="txt" href="rearran1.html#E6:50"><i><font color="Green">E45</font></i></a>, <a class="txt" href="rearran1.html#E7:50"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E10:50_1_1_1_2"/>
<font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="rearran1.html#D1" target="_self">Def1</a>, <a class="txt" href="rearran1.html#E6:50"><i><font color="Green">E45</font></i></a>, <a class="txt" href="rearran1.html#E7:50"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E11:50_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">D</font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">B</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font>
<b>by </b><i><a class="txt" href="rearran1.html#E4:50"><i><font color="Green">E43</font></i></a>, <a class="txt" href="rearran1.html#E10:50"><i><font color="Green">E54</font></i></a>, <a class="txt" href="rearran1.html#E4:50_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="rvsum_1.html#T47">RVSUM_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font>
<b>by </b><i><a class="txt" href="rearran1.html#E5"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="rfunct_3.html#D12">RFUNCT_3:def 12</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E17:50_1"/>
<span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">B</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:50"/>
 <a href="rearran1.html#K4" target="_self">Rlor</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" target="_self">Rlor</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="txt" href="rearran1.html#E5"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div>
