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<b>let </b><font color="Maroon" title="c1">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c2">D</font>, <font color="Maroon" title="c3">C</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c4">F</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">D</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c5">B</font> be   <a href="rearran1.html#NM1" title="REARRAN1:NM.1">RearrangmentGen</a> of <font color="Maroon" title="c3">C</font>;<br/>









<b>assume </b><a NAME="E1:45"/><i><font color="Green" title="E35">A1</font></i>: 
( <font color="Maroon" title="c4">F</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">C</font> )
 ;<br/>

<a NAME="E2:45"/><b>then </b><i><font color="Green" title="E36">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></i>;<br/>
<a NAME="E3:45"/><b>then </b><i><font color="Green" title="E37">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="ref" href="valued_1.html#T3" title="VALUED_1:th.3">VALUED_1:3</a></i>;<br/>
<a NAME="E4:45"/><b>then </b><i><font color="Green" title="E38">A4</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 
<b>by </b><i><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></i>;<br/>
<a NAME="E5:45"/><b>then </b><i><font color="Green" title="E39">A5</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span>,<font color="Maroon" title="c5">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E1:45"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="rearran1.html#T13" target="_self" title="REARRAN1:th.13">Th13</a></i>;<br/>
<a NAME="E6:45"/><b>then </b><i><font color="Green" title="E40">A6</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">B</font></span>)</span> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="ref" href="valued_1.html#T3" title="VALUED_1:th.3">VALUED_1:3</a></i>;<br/>
<a NAME="E7:45"/><i><font color="Green" title="E41">A7</font></i>: 
( <font color="Maroon" title="c4">F</font> <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">|</a> <font color="Maroon" title="c2">D</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">F</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">|</a> <font color="Maroon" title="c2">D</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E2:45"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="rearran1.html#E3:45"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></i>;<br/>
<a NAME="E8:45"/><b>then </b><i><font color="Green" title="E42">A8</font></i>: 
 <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span>,<font color="Maroon" title="c2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <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> <font color="Maroon" title="c2">D</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rearran1.html#E2:45"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="rfunct_3.html#T76" title="RFUNCT_3:th.76">RFUNCT_3:76</a></i>;<br/>
<a NAME="E9:45"/><i><font color="Green" title="E43">A9</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span>,<font color="Maroon" title="c2">D</font></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> <font color="Maroon" title="c2">D</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></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> <font color="Maroon" title="c2">D</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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> <font color="Maroon" title="c2">D</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c1">r</font></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> <font color="Maroon" title="c2">D</font> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E2:45"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="rearran1.html#E3:45"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="rearran1.html#E7:45"><i><font color="Green" title="E41">A7</font></i></a>, <a class="ref" href="finseq_2.html#T69" title="FINSEQ_2:th.69">FINSEQ_2:69</a>, <a class="ref" href="rfunct_3.html#T70" title="RFUNCT_3:th.70">RFUNCT_3:70</a></i>;<br/>
<a NAME="E10:45"/><i><font color="Green" title="E44">A10</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="ref" href="rearran1.html#T1" target="_self" title="REARRAN1:th.1">Th1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">C</font>;<br/><b>assume </b><a NAME="E1:45_1"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span>,<font color="Maroon" title="c5">B</font></span>)</span>
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $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="c5">B</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 );<br/><a NAME="E2:45_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="rearran1.html#T4" target="_self" title="REARRAN1:th.4">Th4</a></i>;<br/><a NAME="E3:45_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c5">B</font>
 ;<br/><a NAME="E4:45_1"/><b>then </b><i><font color="Green" title="E45">A11</font></i>: 
<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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">B</font>
 <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="E5:45_1"/><b>then </b><i><font color="Green" title="E46">A12</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">B</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="c5">B</font> &amp; 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="c5">B</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:45_1"/>
<font color="Maroon" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_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="c5">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="rearran1.html#T3" target="_self" title="REARRAN1:th.3">Th3</a></i>;<br/><a NAME="E7:45_1"/><b>then </b><i><font color="Green" title="E47">A13</font></i>: 
 ex <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>]
 <b>by </b><i><a class="txt" href="rearran1.html#E5:45_1"><i><font color="Green" title="E46">A12</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c7">mi</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:45_1"/><i><font color="Green" title="E48">A14</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">mi</font>] &amp; ( 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" title="c7">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="rearran1.html#E7:45_1"><i><font color="Green" title="E47">A13</font></i></a>);<br/></i><a NAME="E10:45_1"/><i><font color="Green" title="E49">A15</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">mi</font> &amp; <font color="Maroon" title="c7">mi</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="c5">B</font> )
 <b>by </b><i><a class="txt" href="rearran1.html#E9:45_1"><i><font color="Green" title="E48">A14</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:45_1"/><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="c7">mi</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">mi</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c8">m1</font> = <font color="Maroon" title="c7">mi</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</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/><a NAME="E13:45_1"/>
( <font color="Maroon" title="c8">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp; <font color="Maroon" title="c7">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">mi</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c7">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">mi</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="rearran1.html#E10:45_1"><i><font color="Green" title="E49">A15</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E14:45_1"/><b>then </b>
( <font color="Maroon" title="c8">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">mi</font> &amp; <font color="Maroon" title="c8">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">mi</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E15:45_1"/><b>then </b><i><font color="Green" title="E50">A16</font></i>: 
( <font color="Maroon" title="c8">m1</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="c5">B</font> &amp; <font color="Maroon" title="c8">m1</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="c5">B</font> )
 <b>by </b><i><a class="txt" href="rearran1.html#E10:45_1"><i><font color="Green" title="E49">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_1_1"/>
( <font color="Maroon" title="c7">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_1"><b>case </b></a><a NAME="E1:45_1_1_1_1"/>
<font color="Maroon" title="c7">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:45_1_1_1_1"/><b>then </b><i><font color="Green" title="E51">A17</font></i>: 
( <span class="p1">(<span class="default"><a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span>,<font color="Maroon" title="c5">B</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 &amp; <span class="p1">(<span class="default"><a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">B</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 )
 <b>by </b><i><a class="txt" href="rearran1.html#E1:45"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="rearran1.html#E4:45"><i><font color="Green" title="E38">A4</font></i></a>, <a class="txt" href="rearran1.html#E9:45_1"><i><font color="Green" title="E48">A14</font></i></a>, <a class="ref" href="rearran1.html#T15" target="_self" title="REARRAN1:th.15">Th15</a></i>;<br/><a NAME="E3:45_1_1_1_1"/><i><font color="Green" title="E52">A18</font></i>: 
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> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span>,<font color="Maroon" title="c2">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:45"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="rearran1.html#E9:45"><i><font color="Green" title="E43">A9</font></i></a>, <a class="txt" href="rearran1.html#E10:45"><i><font color="Green" title="E44">A10</font></i></a>, <a class="txt" href="rearran1.html#E4:45_1"><i><font color="Green" title="E45">A11</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:45_1_1_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:45"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="rearran1.html#E10:45"><i><font color="Green" title="E44">A10</font></i></a>, <a class="txt" href="rearran1.html#E5:45_1"><i><font color="Green" title="E46">A12</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:45_1_1_1_1"/><b>then </b>
<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> <font color="Maroon" title="c2">D</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span>,<font color="Maroon" title="c5">B</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">c</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">B</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">r</font>
<b>by </b><i><a class="txt" href="rearran1.html#E8:45"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="rearran1.html#E2:45_1_1_1_1"><i><font color="Green" title="E51">A17</font></i></a>, <a class="txt" href="rearran1.html#E3:45_1_1_1_1"><i><font color="Green" title="E52">A18</font></i></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">B</font></span>)</span> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">c</font>
<b>by </b><i><a class="txt" href="rearran1.html#E5:45"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="valued_1.html#T3" title="VALUED_1:th.3">VALUED_1:3</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_2"><b>case </b></a><a NAME="E1:45_1_1_1_2"/>
<font color="Maroon" title="c7">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:45_1_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">mi</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E10:45_1"><i><font color="Green" title="E49">A15</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:45_1_1_1_2"/><b>then </b>
1 <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="c7">mi</font>
 <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="E4:45_1_1_1_2"/><b>then </b><i><font color="Green" title="E51">A19</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E5:45_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">m1</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="c5">B</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E15:45_1"><i><font color="Green" title="E50">A16</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:45_1_1_1_2"/><b>then </b>
not <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m1</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E9:45_1"><i><font color="Green" title="E48">A14</font></i></a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></i>;<br/><a NAME="E7:45_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m1</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E9:45_1"><i><font color="Green" title="E48">A14</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E8:45_1_1_1_2"/><b>then </b><i><font color="Green" title="E52">A20</font></i>: 
( <span class="p1">(<span class="default"><a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span>,<font color="Maroon" title="c5">B</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">B</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="rearran1.html#E1:45"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="rearran1.html#E4:45"><i><font color="Green" title="E38">A4</font></i></a>, <a class="txt" href="rearran1.html#E15:45_1"><i><font color="Green" title="E50">A16</font></i></a>, <a class="txt" href="rearran1.html#E4:45_1_1_1_2"><i><font color="Green" title="E51">A19</font></i></a>, <a class="ref" href="rearran1.html#T15" target="_self" title="REARRAN1:th.15">Th15</a></i>;<br/><a NAME="E9:45_1_1_1_2"/><i><font color="Green" title="E53">A21</font></i>: 
<font color="Maroon" title="c8">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span>,<font color="Maroon" title="c2">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:45"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="rearran1.html#E9:45"><i><font color="Green" title="E43">A9</font></i></a>, <a class="txt" href="rearran1.html#E10:45"><i><font color="Green" title="E44">A10</font></i></a>, <a class="txt" href="rearran1.html#E9:45_1"><i><font color="Green" title="E48">A14</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E10:45_1_1_1_2"/>
<font color="Maroon" title="c8">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:45"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="rearran1.html#E10:45"><i><font color="Green" title="E44">A10</font></i></a>, <a class="txt" href="rearran1.html#E9:45_1"><i><font color="Green" title="E48">A14</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E11:45_1_1_1_2"/><b>then </b>
<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> <font color="Maroon" title="c2">D</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span>,<font color="Maroon" title="c5">B</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">c</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">B</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">r</font>
<b>by </b><i><a class="txt" href="rearran1.html#E8:45"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="rearran1.html#E8:45_1_1_1_2"><i><font color="Green" title="E52">A20</font></i></a>, <a class="txt" href="rearran1.html#E9:45_1_1_1_2"><i><font color="Green" title="E53">A21</font></i></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">B</font></span>)</span> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">c</font>
<b>by </b><i><a class="txt" href="rearran1.html#E5:45"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="valued_1.html#T3" title="VALUED_1:th.3">VALUED_1:3</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E17:45_1"/>
<span class="p1">(<span class="default"><a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span>,<font color="Maroon" title="c5">B</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">B</font></span>)</span> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">c</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:45"/>
 <a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font></span>)</span>,<font color="Maroon" title="c5">B</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">B</font></span>)</span> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E5:45"><i><font color="Green" title="E39">A5</font></i></a>, <a class="txt" href="rearran1.html#E6:45"><i><font color="Green" title="E40">A6</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>;<br/>


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