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<b>let </b><font color="Maroon" title="c1">D</font>, <font color="Maroon" title="c2">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="c3">F</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">D</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c4">B</font> be   <a href="rearran1.html#NM1" title="REARRAN1:NM.1">RearrangmentGen</a> of <font color="Maroon" title="c2">C</font>;<br/>







<b>set </b><font color="Maroon" title="c5">fd</font> =  <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font>;<br/>
<b>set </b><font color="Maroon" title="c6">p</font> =  <a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">B</font>;<br/>
<b>set </b><font color="Maroon" title="c7">mf</font> =  <a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">b</font> =  <a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font>;<br/>
<b>set </b><font color="Maroon" title="c9">h</font> =  <a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span>,<font color="Maroon" title="c2">C</font>;<br/>
<b>assume </b><a NAME="E1:36"/><i><font color="Green" title="E32">A1</font></i>: 
( <font color="Maroon" title="c3">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">C</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="c1">D</font> )
 ;<br/>

<a NAME="E2:36"/><b>then </b><i><font color="Green" title="E33">A2</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rearran1.html#T23" target="_self" title="REARRAN1:th.23">Th23</a></i>;<br/>
<a NAME="E3:36"/><i><font color="Green" title="E34">A3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span>,<font color="Maroon" title="c2">C</font></span>)</span> &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#K16" title="RFUNCT_3:func.16">CHI</a> <span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span>,<font color="Maroon" title="c2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E1:36"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="rearran1.html#T12" target="_self" title="REARRAN1:th.12">Th12</a>, <a class="ref" href="rfinseq.html#D3" title="RFINSEQ:def.3">RFINSEQ:def 3</a>, <a class="ref" href="rfunct_3.html#D6" title="RFUNCT_3:def.6">RFUNCT_3:def 6</a></i>;<br/>
<a NAME="E4:36"/><i><font color="Green" title="E35">A4</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#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">D</font> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E1:36"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="rearran1.html#T21" target="_self" title="REARRAN1:th.21">Th21</a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">p</font> =  <a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">B</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">C</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="finset_1.html#T29" title="FINSET_1:th.29">FINSET_1:29</a></i>;<br/>
<a NAME="E6:36"/><i><font color="Green" title="E36">A5</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1"/>
(  not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_1"/>
not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="rearran1.html#E2:36"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="rearran1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><b>hence </b><a NAME="E3:36_1_1_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2"/><i><font color="Green" title="E37">A6</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> );<br/><a NAME="E2:36_1_1_1_2"/><i><font color="Green" title="E38">A100</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#E1:36_1_1_1_2"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/><a NAME="E3:36_1_1_1_2"/><i><font color="Green" title="E39">A8</font></i>: 
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="Olive" title="b1">n</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> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">mi</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:36_1_1_1_2"/><i><font color="Green" title="E40">A9</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">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="c12">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#E2:36_1_1_1_2"><i><font color="Green" title="E38">A100</font></i></a>);<br/></i><b>consider </b><font color="Maroon" title="c13">ma</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:36_1_1_1_2"/><i><font color="Green" title="E41">A10</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">ma</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="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="rearran1.html#E3:36_1_1_1_2"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2"><i><font color="Green" title="E38">A100</font></i></a>);<br/></i><a NAME="E8:36_1_1_1_2"/><i><font color="Green" title="E42">A11</font></i>: 
<font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green" title="E41">A10</font></i></a></i>;<br/><a NAME="E9:36_1_1_1_2"/><i><font color="Green" title="E43">A12</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">mi</font> &amp; <font color="Maroon" title="c12">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> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</font> &amp; <font color="Maroon" title="c13">ma</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> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> )
 <b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:36_1_1_1_2"/><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="c12">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="c12">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="c14">m1</font> = <font color="Maroon" title="c12">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/><b>reconsider </b><font color="Maroon" title="c15">r</font> = <font color="Maroon" title="c11">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green" title="E40">A9</font></i></a></i>;<br/><a NAME="E13:36_1_1_1_2"/>
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> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green" title="E43">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E14:36_1_1_1_2"/><b>then </b><i><font color="Green" title="E44">A13</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#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span>
 <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="E15:36_1_1_1_2"/>
<font color="Maroon" title="c14">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">mi</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T45" title="XREAL_1:th.45">XREAL_1:45</a></i>;<br/><a NAME="E16:36_1_1_1_2"/><b>then </b><i><font color="Green" title="E45">A14</font></i>: 
( <font color="Maroon" title="c14">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</font> &amp; <font color="Maroon" title="c14">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">mi</font> )
 <b>by </b><i><a class="txt" href="rearran1.html#E8:36_1_1_1_2"><i><font color="Green" title="E42">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E17:36_1_1_1_2"/><b>then </b><i><font color="Green" title="E46">A15</font></i>: 
(  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c14">m1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">ma</font> &amp;  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">ma</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/><a NAME="E18:36_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">ma</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:36_1_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">ma</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:36_1_1_1_2_1_1"/><i><font color="Green" title="E47">A16</font></i>: 
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">ma</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c14">m1</font></span>)</span>
 ;<br/>

<a NAME="E2:36_1_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E48">A17</font></i>: 
( <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">ma</font> &amp; not <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c14">m1</font> &amp;  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">ma</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">l</font> = <font color="Maroon" title="c16">y</font> 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="txt" href="rearran1.html#E1:36_1_1_1_2_1_1"><i><font color="Green" title="E47">A16</font></i></a></i>;<br/>
<a NAME="E4:36_1_1_1_2_1_1"/><i><font color="Green" title="E49">A18</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">l</font> &amp; <font color="Maroon" title="c17">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</font> &amp; ( <font color="Maroon" title="c17">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 or <font color="Maroon" title="c14">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">l</font> ) )
 
<b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2_1_1"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:36_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<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="E6:36_1_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E50">A19</font></i>: 
<font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">l</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="E7:36_1_1_1_2_1_1"/>
<font color="Maroon" title="c17">l</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> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_1_1"><i><font color="Green" title="E49">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:36_1_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E51">A20</font></i>: 
<font color="Maroon" title="c17">l</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> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_1_1"><i><font color="Green" title="E49">A18</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">o</font> = <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">l</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_1_1_1_1"/>
( <font color="Maroon" title="c17">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ma</font> or <font color="Maroon" title="c17">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">ma</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c17">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ma</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c18">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:36_1_1_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E8:36_1_1_1_2_1_1"><i><font color="Green" title="E51">A20</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c17">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">ma</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c17">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c13">ma</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_1_1"><i><font color="Green" title="E49">A18</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="E3:36_1_1_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E52">A21</font></i>: 
<font color="Maroon" title="c18">o</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">r</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="txt" href="rearran1.html#E8:36_1_1_1_2_1_1"><i><font color="Green" title="E51">A20</font></i></a>, <a class="ref" href="rfinseq.html#T32" title="RFINSEQ:th.32">RFINSEQ:32</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c17">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">mi</font> or <font color="Maroon" title="c17">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">mi</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_1_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c17">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">mi</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c18">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green" title="E40">A9</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:36_1_1_1_2_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E8:36_1_1_1_2_1_1"><i><font color="Green" title="E51">A20</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_1_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c17">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">mi</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">l</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E6:36_1_1_1_2_1_1"><i><font color="Green" title="E50">A19</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="E3:36_1_1_1_2_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c15">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c18">o</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="rearran1.html#E8:36_1_1_1_2_1_1"><i><font color="Green" title="E51">A20</font></i></a>, <a class="ref" href="rfinseq.html#T32" title="RFINSEQ:th.32">RFINSEQ:32</a></i>;<br/><a NAME="E4:36_1_1_1_2_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c15">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">o</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36_1_1_1_2_1_1_1_1_2"><i><font color="Green" title="E52">A21</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:36_1_1_1_2_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c18">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:36_1_1_1_2_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E8:36_1_1_1_2_1_1"><i><font color="Green" title="E51">A20</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:36_1_1_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:36_1_1_1_2_1_1"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c16">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:36_1_1_1_2_1"/><i><font color="Green" title="E47">A22</font></i>: 
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E3:36_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E48">A23</font></i>: 
( <font color="Maroon" title="c16">y</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> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">l</font> = <font color="Maroon" title="c16">y</font> 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="txt" href="rearran1.html#E2:36_1_1_1_2_1"><i><font color="Green" title="E47">A22</font></i></a></i>;<br/>
<a NAME="E5:36_1_1_1_2_1"/><i><font color="Green" title="E49">A24</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font>
 
<b>by </b><i><a class="txt" href="rearran1.html#E3:36_1_1_1_2_1"><i><font color="Green" title="E48">A23</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:36_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">l</font>
 
<b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2_1"><i><font color="Green" title="E48">A23</font></i></a></i>;<br/>
<a NAME="E7:36_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<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="E8:36_1_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c14">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">l</font> or <font color="Maroon" title="c17">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 
<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="E9:36_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E50">A25</font></i>: 
not <font color="Maroon" title="c17">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c14">m1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:36_1_1_1_2_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">l</font> &amp; <font color="Maroon" title="c17">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</font> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2_1"><i><font color="Green" title="E48">A23</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2_1"><i><font color="Green" title="E49">A24</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:36_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c17">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">ma</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E12:36_1_1_1_2_1"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">ma</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c14">m1</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2_1"><i><font color="Green" title="E50">A25</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><b>then </b><i><font color="Green" title="E47">A26</font></i>:  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">ma</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c14">m1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rearran1.html#E17:36_1_1_1_2"><i><font color="Green" title="E46">A15</font></i></a>, <a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></i>
<br/>.= 
<font color="Maroon" title="c13">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c14">m1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a></i>
<br/>.= 
<font color="Maroon" title="c13">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">m1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1"/>
( <font color="Maroon" title="c12">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c12">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="36_1_1_1_2_3_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1"/><i><font color="Green" title="E48">A27</font></i>: 
<font color="Maroon" title="c12">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_1"/>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:36_1_1_1_2_3_1_1_1"/>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ma</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:36_1_1_1_2_3_1_1_1_1"/><i><font color="Green" title="E49">A28</font></i>: 
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ma</font>
 ;<br/>

<a NAME="E2:36_1_1_1_2_3_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ma</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="rearran1.html#E6:36"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="txt" href="rearran1.html#E14"><i><font color="Green" title="E14">Lm5</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">cy</font> = <font color="Maroon" title="c16">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C</font> <b>by </b><i><a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_1_1_1"><i><font color="Green" title="E49">A28</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <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>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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</font> implies for <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> );<br/>
<a NAME="E4:36_1_1_1_2_3_1_1_1_1"/><i><font color="Green" title="E50">A29</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<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="E5:36_1_1_1_2_3_1_1_1_1"/><i><font color="Green" title="E51">A30</font></i>: 
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>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">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> ;<br/>

<b>assume </b><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1"/><i><font color="Green" title="E52">A31</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c18">n</font>]
 ;<br/>

<b>assume </b><a NAME="E2:36_1_1_1_2_3_1_1_1_1_1"/><i><font color="Green" title="E53">A32</font></i>: 
( <font color="Maroon" title="c18">n</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="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> &amp; <font color="Maroon" title="c18">n</font> <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="c13">ma</font> )
 ;<br/>

<a NAME="E3:36_1_1_1_2_3_1_1_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">n</font> <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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> &amp; <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E4:36_1_1_1_2_3_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E54">A33</font></i>: 
( <font color="Maroon" title="c18">n</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> &amp; <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</font> &amp; <font color="Maroon" title="c18">n</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> <span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp; <font color="Maroon" title="c18">n</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_1_1_1_1"><i><font color="Green" title="E53">A32</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#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<b>let </b><font color="Maroon" title="c19">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C</font>;<br/>

<b>assume </b><a NAME="E5:36_1_1_1_2_3_1_1_1_1_1"/><i><font color="Green" title="E55">A34</font></i>: 
<font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c18">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_1"><i><font color="Green" title="E48">A27</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_1_1_1_1"><i><font color="Green" title="E55">A34</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self" title="REARRAN1:th.22">Th22</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c18">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_1_1_1_1_1_1_2"/><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> <font color="Maroon" title="c18">n</font>
 ;<br/><a NAME="E3:36_1_1_1_2_3_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E56">A35</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> <font color="Maroon" title="c18">n</font> &amp; <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> <font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E4:36_1_1_1_2_3_1_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">n</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#E4:36_1_1_1_2_3_1_1_1_1_1"><i><font color="Green" title="E54">A33</font></i></a>, <a class="ref" href="rearran1.html#D2" target="_self" title="REARRAN1:def.2">Def2</a></i>;<br/><b>then </b><i><font color="Green" title="E57">A36</font></i>: <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">n</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="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1"/>
( <font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">n</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"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> or <font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> )
 </span><b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_1_1_1_1"><i><font color="Green" title="E55">A34</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_1_1_1_1_1_1_2"><i><font color="Green" title="E57">A36</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">n</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"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E58">A37</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">n</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#E1:36"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_1_1_1_1"><i><font color="Green" title="E54">A33</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2_3_1_1_1_1_1_1_1_2"><i><font color="Green" title="E56">A35</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self" title="REARRAN1:th.22">Th22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1"/>
( <font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ma</font> or <font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">ma</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_1"/>
<font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ma</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1"><i><font color="Green" title="E58">A37</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_1"/>
<font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/>
<font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">ma</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c18">n</font> <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="c13">ma</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_1_1_1_1"><i><font color="Green" title="E53">A32</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="E3:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E59">A38</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">r</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="rearran1.html#E6:36"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_1_1_1_1"><i><font color="Green" title="E53">A32</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1"><i><font color="Green" title="E58">A37</font></i></a>, <a class="ref" href="rfinseq.html#T32" title="RFINSEQ:th.32">RFINSEQ:32</a></i>;<br/><a NAME="E4:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/>
<font color="Maroon" title="c15">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">c</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="rearran1.html#E6:36"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_1"><i><font color="Green" title="E48">A27</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_1_1_1_1"><i><font color="Green" title="E53">A32</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2_3_1_1_1_1_1_1_1_2"><i><font color="Green" title="E56">A35</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1"><i><font color="Green" title="E58">A37</font></i></a>, <a class="ref" href="rfinseq.html#T32" title="RFINSEQ:th.32">RFINSEQ:32</a></i>;<br/><a NAME="E5:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c15">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">c</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"><i><font color="Green" title="E59">A38</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E6:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/>
<font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_2"/>
<font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_1_1_1_2_3_1_1_1_1_1_1_1_2_2_1_2"/>
<font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_1_1_1_1"><i><font color="Green" title="E52">A31</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_1_1_1_1"><i><font color="Green" title="E54">A33</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2_3_1_1_1_1_1_1_1_2"><i><font color="Green" title="E56">A35</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:36_1_1_1_2_3_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:36_1_1_1_2_3_1_1_1_1_1"/>
<font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:36_1_1_1_2_3_1_1_1_1"/><i><font color="Green" title="E52">A39</font></i>: 
<font color="Maroon" title="c13">ma</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E7:36_1_1_1_2_3_1_1_1_1"/>
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>2</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_1_1_1"><i><font color="Green" title="E50">A29</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_1_1_1"><i><font color="Green" title="E51">A30</font></i></a>);<br/></i>
<a NAME="E8:36_1_1_1_2_3_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">cy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_1_1_1"><i><font color="Green" title="E49">A28</font></i></a>, <a class="txt" href="rearran1.html#E6:36_1_1_1_2_3_1_1_1_1"><i><font color="Green" title="E52">A39</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:36_1_1_1_2_3_1_1_1_1"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c16">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:36_1_1_1_2_3_1_1_1"/><i><font color="Green" title="E49">A40</font></i>: 
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c17">cy</font> = <font color="Maroon" title="c16">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C</font> ;<br/>
<b>assume </b><a NAME="E4:36_1_1_1_2_3_1_1_1"/><i><font color="Green" title="E50">A41</font></i>: 
not <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ma</font>
 ;<br/>

<a NAME="E5:36_1_1_1_2_3_1_1_1"/>
( <font color="Maroon" title="c17">cy</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> <font color="Maroon" title="c10">p</font> &amp; <font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">cy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_1_1"><i><font color="Green" title="E49">A40</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E6:36_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green" title="E51">A42</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">cy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> &amp; <font color="Maroon" title="c13">ma</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> $1 &amp; <font color="Maroon" title="c17">cy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 );<br/>
<a NAME="E7:36_1_1_1_2_3_1_1_1"/><i><font color="Green" title="E52">A43</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>2</sub></font>[<font color="Olive" title="b1">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_2"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c18">n</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span>;<br/>

<a NAME="E1:36_1_1_1_2_3_1_1_1_2"/><i><font color="Green" title="E53">A44</font></i>: 
<span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="ref" href="rearran1.html#T3" target="_self" title="REARRAN1:th.3">Th3</a></i>;<br/>
<a NAME="E2:36_1_1_1_2_3_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <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="E3:36_1_1_1_2_3_1_1_1_2"/><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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span>
 
;<br/>
<a NAME="E4:36_1_1_1_2_3_1_1_1_2"/><b>then </b>
<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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>hence </b><a NAME="E5:36_1_1_1_2_3_1_1_1_2"/>
<font color="Maroon" title="c18">n</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_1_1_2_3_1_1_1_2_1"/>
<font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</font>
 ;<br/><a NAME="E2:36_1_1_1_2_3_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c13">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green" title="E43">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c17">cy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ma</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_1_1_2"><i><font color="Green" title="E53">A44</font></i></a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_1_1"><i><font color="Green" title="E50">A41</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:36_1_1_1_2_3_1_1_1_2"/>
( <font color="Maroon" title="c13">ma</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">n</font> &amp; <font color="Maroon" title="c17">cy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> )
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_1_1_2"><i><font color="Green" title="E53">A44</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c18">mm</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:36_1_1_1_2_3_1_1_1"/><i><font color="Green" title="E53">A45</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c18">mm</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>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c18">mm</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#E7:36_1_1_1_2_3_1_1_1"><i><font color="Green" title="E52">A43</font></i></a>);<br/></i>
<a NAME="E10:36_1_1_1_2_3_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">mm</font>
 
<b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2_3_1_1_1"><i><font color="Green" title="E53">A45</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:36_1_1_1_2_3_1_1_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="c18">mm</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="c18">mm</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="c19">1m</font> = <font color="Maroon" title="c18">mm</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="E13:36_1_1_1_2_3_1_1_1"/>
( <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">mm</font> &amp; <font color="Maroon" title="c18">mm</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">mm</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c18">mm</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">mm</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2_3_1_1_1"><i><font color="Green" title="E53">A45</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E14:36_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green" title="E54">A46</font></i>: 
( <font color="Maroon" title="c13">ma</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">1m</font> &amp; <font color="Maroon" title="c18">mm</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> &amp; <font color="Maroon" title="c19">1m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">mm</font> &amp; <font color="Maroon" title="c19">1m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">mm</font> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2_3_1_1_1"><i><font color="Green" title="E53">A45</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E15:36_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green" title="E55">A47</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">1m</font> &amp; <font color="Maroon" title="c19">1m</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> &amp; <font color="Maroon" title="c19">1m</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green" title="E43">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:36_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green" title="E56">A48</font></i>: 
( <font color="Maroon" title="c19">1m</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> &amp; <font color="Maroon" title="c19">1m</font> <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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E17:36_1_1_1_2_3_1_1_1"/><i><font color="Green" title="E57">A49</font></i>: 
<font color="Maroon" title="c18">mm</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> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="rearran1.html#E9:36_1_1_1_2_3_1_1_1"><i><font color="Green" title="E53">A45</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_1_1_3_1"/>
( <font color="Maroon" title="c13">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">1m</font> or <font color="Maroon" title="c13">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">1m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_3_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1_1_3_1_1"/>
<font color="Maroon" title="c13">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">1m</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c17">cy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">1m</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"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">1m</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_1_1"><i><font color="Green" title="E50">A41</font></i></a>, <a class="txt" href="rearran1.html#E9:36_1_1_1_2_3_1_1_1"><i><font color="Green" title="E53">A45</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="E3:36_1_1_1_2_3_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">cy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">mm</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="rearran1.html#E15:36_1_1_1_2_3_1_1_1"><i><font color="Green" title="E55">A47</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self" title="REARRAN1:th.22">Th22</a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_1_1_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="txt" href="rearran1.html#E6:36_1_1_1_2_3_1_1_1"><i><font color="Green" title="E51">A42</font></i></a>, <a class="txt" href="rearran1.html#E9:36_1_1_1_2_3_1_1_1"><i><font color="Green" title="E53">A45</font></i></a>, <a class="txt" href="rearran1.html#E17:36_1_1_1_2_3_1_1_1"><i><font color="Green" title="E57">A49</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_3_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_1_1_3_1_2"/>
<font color="Maroon" title="c13">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">1m</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_1_1_3_1_2"/><b>then </b><i><font color="Green" title="E58">A50</font></i>: 
<font color="Maroon" title="c13">ma</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">1m</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E14:36_1_1_1_2_3_1_1_1"><i><font color="Green" title="E54">A46</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_1_1_3_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_1_1_2_3_1_1_1_3_1_2_1"/>
<font color="Maroon" title="c17">cy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">1m</font>
 ;<br/><a NAME="E2:36_1_1_1_2_3_1_1_1_3_1_2_1"/><b>then </b>
<font color="Maroon" title="c18">mm</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">1m</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2_3_1_1_1"><i><font color="Green" title="E53">A45</font></i></a>, <a class="txt" href="rearran1.html#E16:36_1_1_1_2_3_1_1_1"><i><font color="Green" title="E56">A48</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_1_1_3_1_2"><i><font color="Green" title="E58">A50</font></i></a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_1_1_3_1_2_1"/><b>then </b>
<font color="Maroon" title="c18">mm</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">1m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_1_1_3_1_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E4:36_1_1_1_2_3_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c17">cy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">1m</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"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">1m</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2_3_1_1_1"><i><font color="Green" title="E53">A45</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="E5:36_1_1_1_2_3_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">cy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">mm</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="rearran1.html#E15:36_1_1_1_2_3_1_1_1"><i><font color="Green" title="E55">A47</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self" title="REARRAN1:th.22">Th22</a></i>;<br/><b>hence </b><a NAME="E6:36_1_1_1_2_3_1_1_1_3_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="txt" href="rearran1.html#E6:36_1_1_1_2_3_1_1_1"><i><font color="Green" title="E51">A42</font></i></a>, <a class="txt" href="rearran1.html#E9:36_1_1_1_2_3_1_1_1"><i><font color="Green" title="E53">A45</font></i></a>, <a class="txt" href="rearran1.html#E17:36_1_1_1_2_3_1_1_1"><i><font color="Green" title="E57">A49</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E19:36_1_1_1_2_3_1_1_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:36_1_1_1_2_3_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="rearran1.html#E19:36_1_1_1_2"><i><font color="Green" title="E47">A26</font></i></a>, <a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_1"><i><font color="Green" title="E48">A27</font></i></a>, <a class="ref" href="rearran1.html#D1" target="_self" title="REARRAN1:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2"/><i><font color="Green" title="E48">A51</font></i>: 
<font color="Maroon" title="c12">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">mi</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green" title="E43">A12</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="E3:36_1_1_1_2_3_1_2"/><b>then </b><i><font color="Green" title="E49">A52</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">m1</font> &amp; <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> )
 <b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="rearran1.html#E16:36_1_1_1_2"><i><font color="Green" title="E45">A14</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:36_1_1_1_2_3_1_2"/><b>then </b><i><font color="Green" title="E50">A53</font></i>: 
<font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:36_1_1_1_2_3_1_2"/><b>then </b><i><font color="Green" title="E51">A54</font></i>: 
<span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ma</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="rearran1.html#E6:36"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="txt" href="rearran1.html#E16:36_1_1_1_2"><i><font color="Green" title="E45">A14</font></i></a>, <a class="ref" href="rearran1.html#T2" target="_self" title="REARRAN1:th.2">Th2</a></i>;<br/><a NAME="E6:36_1_1_1_2_3_1_2"/>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ma</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">C</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="rearran1.html#E6:36"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="txt" href="rearran1.html#E14"><i><font color="Green" title="E14">Lm5</font></i></a></i>;<br/><a NAME="E7:36_1_1_1_2_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ma</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c16">bma</font> = <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ma</font>, <font color="Maroon" title="c17">bm1</font> = <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m1</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_2"><i><font color="Green" title="E51">A54</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/><a NAME="E9:36_1_1_1_2_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ma</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:36_1_1_1_2_3_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ma</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:36_1_1_1_2_3_1_2_1_1"/><i><font color="Green" title="E52">A55</font></i>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ma</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m1</font></span>)</span>
 ;<br/>

<a NAME="E2:36_1_1_1_2_3_1_2_1_1"/><b>then </b><i><font color="Green" title="E53">A56</font></i>: 
( <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ma</font> &amp; not <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m1</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:36_1_1_1_2_3_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ma</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="rearran1.html#E6:36"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="txt" href="rearran1.html#E14"><i><font color="Green" title="E14">Lm5</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c19">cy</font> = <font color="Maroon" title="c18">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C</font> <b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_1"><i><font color="Green" title="E53">A56</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <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>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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> &amp; <font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</font> implies for <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> );<br/>
<a NAME="E5:36_1_1_1_2_3_1_2_1_1"/><i><font color="Green" title="E54">A57</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<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="E6:36_1_1_1_2_3_1_2_1_1"/><i><font color="Green" title="E55">A58</font></i>: 
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>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">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> ;<br/>

<b>assume </b><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1"/><i><font color="Green" title="E56">A59</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c20">n</font>]
 ;<br/>

<b>assume </b><a NAME="E2:36_1_1_1_2_3_1_2_1_1_1"/><i><font color="Green" title="E57">A60</font></i>: 
( <font color="Maroon" title="c20">n</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="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> &amp; <font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">n</font> <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="c13">ma</font> )
 ;<br/>

<a NAME="E3:36_1_1_1_2_3_1_2_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">n</font> <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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> &amp; <font color="Maroon" title="c20">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E4:36_1_1_1_2_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E58">A61</font></i>: 
( <font color="Maroon" title="c20">n</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> &amp; <font color="Maroon" title="c20">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</font> &amp; <font color="Maroon" title="c20">n</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> <span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp; <font color="Maroon" title="c20">n</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green" title="E57">A60</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#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<b>let </b><font color="Maroon" title="c21">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C</font>;<br/>

<b>assume </b><a NAME="E5:36_1_1_1_2_3_1_2_1_1_1"/><i><font color="Green" title="E59">A62</font></i>: 
<font color="Maroon" title="c21">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">n</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"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m1</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1"/>
( <font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">mi</font> or <font color="Maroon" title="c12">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1_1"/><i><font color="Green" title="E60">A63</font></i>: 
<font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">mi</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">n</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#E1:36"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2_3_1_2"><i><font color="Green" title="E49">A52</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green" title="E58">A61</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green" title="E59">A62</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self" title="REARRAN1:th.22">Th22</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_2_1_1_1_1_1_1"><i><font color="Green" title="E60">A63</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c21">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c12">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E60">A64</font></i>: 
( <font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green" title="E57">A60</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">n</font> &amp; <font color="Maroon" title="c20">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</font> )
 <b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green" title="E57">A60</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:36_1_1_1_2_3_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E61">A65</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">n</font> &amp; <font color="Maroon" title="c20">n</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> )
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green" title="E43">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:36_1_1_1_2_3_1_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c20">n</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <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="E6:36_1_1_1_2_3_1_2_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">n</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#E2:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green" title="E57">A60</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_1_1_1_1_2"><i><font color="Green" title="E60">A64</font></i></a>, <a class="ref" href="rearran1.html#T2" target="_self" title="REARRAN1:th.2">Th2</a></i>;<br/><b>then </b><i><font color="Green" title="E62">A66</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">n</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"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m1</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c20">n</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="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m1</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c20">n</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="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T42" title="XBOOLE_1:th.42">XBOOLE_1:42</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1"/>
( <font color="Maroon" title="c21">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">n</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="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> or <font color="Maroon" title="c21">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green" title="E59">A62</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2_3_1_2_1_1_1_1_1_2"><i><font color="Green" title="E62">A66</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c21">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">n</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="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m1</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c21">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">n</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"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E63">A67</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">n</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#E1:36"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green" title="E58">A61</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_1_1_1_1_2"><i><font color="Green" title="E61">A65</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self" title="REARRAN1:th.22">Th22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1"/>
( <font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ma</font> or <font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">ma</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_1"/>
<font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ma</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green" title="E63">A67</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_1"/>
<font color="Maroon" title="c21">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/>
<font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">ma</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c20">n</font> <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="c13">ma</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green" title="E57">A60</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="E3:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E64">A68</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">r</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="rearran1.html#E6:36"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green" title="E57">A60</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green" title="E63">A67</font></i></a>, <a class="ref" href="rfinseq.html#T32" title="RFINSEQ:th.32">RFINSEQ:32</a></i>;<br/><a NAME="E4:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/>
<font color="Maroon" title="c15">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c21">c</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="rearran1.html#E6:36"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green" title="E57">A60</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_1_1_1_1_2"><i><font color="Green" title="E60">A64</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green" title="E63">A67</font></i></a>, <a class="ref" href="rfinseq.html#T32" title="RFINSEQ:th.32">RFINSEQ:32</a></i>;<br/><a NAME="E5:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c15">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c21">c</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"><i><font color="Green" title="E64">A68</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E6:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/>
<font color="Maroon" title="c21">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c21">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_2"/>
<font color="Maroon" title="c21">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m1</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_1_1_1_2_3_1_2_1_1_1_1_1_2_2_1_2"/>
<font color="Maroon" title="c21">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green" title="E56">A59</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_1_1"><i><font color="Green" title="E57">A60</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_1_1_1_1_2"><i><font color="Green" title="E60">A64</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_1_1_1_1_2"><i><font color="Green" title="E61">A65</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:36_1_1_1_2_3_1_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c21">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:36_1_1_1_2_3_1_2_1_1_1"/>
<font color="Maroon" title="c21">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:36_1_1_1_2_3_1_2_1_1"/><i><font color="Green" title="E56">A69</font></i>: 
<font color="Maroon" title="c13">ma</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:36_1_1_1_2_3_1_2_1_1"/>
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>2</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_2_1_1"><i><font color="Green" title="E54">A57</font></i></a>, <a class="txt" href="rearran1.html#E6:36_1_1_1_2_3_1_2_1_1"><i><font color="Green" title="E55">A58</font></i></a>);<br/></i>
<a NAME="E9:36_1_1_1_2_3_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c19">cy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="rearran1.html#E8:36_1_1_1_2"><i><font color="Green" title="E42">A11</font></i></a>, <a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_2_1_1"><i><font color="Green" title="E52">A55</font></i></a>, <a class="txt" href="rearran1.html#E7:36_1_1_1_2_3_1_2_1_1"><i><font color="Green" title="E56">A69</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:36_1_1_1_2_3_1_2_1_1"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c18">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:36_1_1_1_2_3_1_2_1"/><i><font color="Green" title="E52">A70</font></i>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c19">cy</font> = <font color="Maroon" title="c18">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C</font> ;<br/>
<a NAME="E4:36_1_1_1_2_3_1_2_1"/>
( <font color="Maroon" title="c19">cy</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> <font color="Maroon" title="c10">p</font> &amp; <font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">cy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1"><i><font color="Green" title="E52">A70</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E5:36_1_1_1_2_3_1_2_1"/><b>then </b><i><font color="Green" title="E53">A71</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">cy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>assume </b><a NAME="E6:36_1_1_1_2_3_1_2_1"/><i><font color="Green" title="E54">A72</font></i>: 
not <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ma</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m1</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1"/>
(  not <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ma</font> or <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m1</font> )
 </span><b>by </b><i><a class="txt" href="rearran1.html#E6:36_1_1_1_2_3_1_2_1"><i><font color="Green" title="E54">A72</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_1"/><i><font color="Green" title="E55">A73</font></i>: 
not <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ma</font>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> &amp; <font color="Maroon" title="c13">ma</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> $1 &amp; <font color="Maroon" title="c19">cy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 );<br/><a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_1"/><i><font color="Green" title="E56">A74</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>2</sub></font>[<font color="Olive" title="b1">n</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c20">n</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span>;<br/>

<a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_1_1"/><i><font color="Green" title="E57">A75</font></i>: 
<span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="ref" href="rearran1.html#T3" target="_self" title="REARRAN1:th.3">Th3</a></i>;<br/>
<a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <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="E3:36_1_1_1_2_3_1_2_1_2_1_1_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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span>
 
;<br/>
<a NAME="E4:36_1_1_1_2_3_1_2_1_2_1_1_1"/><b>then </b>
<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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>hence </b><a NAME="E5:36_1_1_1_2_3_1_2_1_2_1_1_1"/>
<font color="Maroon" title="c20">n</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c20">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</font>
 ;<br/><a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green" title="E43">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c19">cy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ma</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_2_1_2_1_1_1"><i><font color="Green" title="E57">A75</font></i></a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_2_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green" title="E55">A73</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:36_1_1_1_2_3_1_2_1_2_1_1_1"/>
( <font color="Maroon" title="c13">ma</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c20">n</font> &amp; <font color="Maroon" title="c19">cy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font> )
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_2_1_2_1_1_1"><i><font color="Green" title="E57">A75</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c20">mm</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:36_1_1_1_2_3_1_2_1_2_1_1"/><i><font color="Green" title="E57">A76</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c20">mm</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>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c20">mm</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#E2:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green" title="E56">A74</font></i></a>);<br/></i><a NAME="E5:36_1_1_1_2_3_1_2_1_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">mm</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green" title="E57">A76</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:36_1_1_1_2_3_1_2_1_2_1_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="c20">mm</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="c20">mm</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="c21">1m</font> = <font color="Maroon" title="c20">mm</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="E8:36_1_1_1_2_3_1_2_1_2_1_1"/>
( <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">mm</font> &amp; <font color="Maroon" title="c20">mm</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">mm</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c20">mm</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c20">mm</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green" title="E57">A76</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E9:36_1_1_1_2_3_1_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E58">A77</font></i>: 
( <font color="Maroon" title="c13">ma</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">1m</font> &amp; <font color="Maroon" title="c20">mm</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> &amp; <font color="Maroon" title="c21">1m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">mm</font> &amp; <font color="Maroon" title="c21">1m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c20">mm</font> )
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green" title="E57">A76</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E10:36_1_1_1_2_3_1_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E59">A78</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">1m</font> &amp; <font color="Maroon" title="c21">1m</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> &amp; <font color="Maroon" title="c21">1m</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> )
 <b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green" title="E43">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E11:36_1_1_1_2_3_1_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E60">A79</font></i>: 
( <font color="Maroon" title="c21">1m</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> &amp; <font color="Maroon" title="c21">1m</font> <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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E12:36_1_1_1_2_3_1_2_1_2_1_1"/><i><font color="Green" title="E61">A80</font></i>: 
<font color="Maroon" title="c20">mm</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> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green" title="E57">A76</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_1_2_1"/>
( <font color="Maroon" title="c13">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">1m</font> or <font color="Maroon" title="c13">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">1m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_1_2_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_1_2_1_1"/>
<font color="Maroon" title="c13">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">1m</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c19">cy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">1m</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"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">1m</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green" title="E55">A73</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green" title="E57">A76</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="E3:36_1_1_1_2_3_1_2_1_2_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">cy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c20">mm</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="rearran1.html#E10:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green" title="E59">A78</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self" title="REARRAN1:th.22">Th22</a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_2_1_2_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_2_1"><i><font color="Green" title="E53">A71</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green" title="E57">A76</font></i></a>, <a class="txt" href="rearran1.html#E12:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green" title="E61">A80</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_1_2_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2"/>
<font color="Maroon" title="c13">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">1m</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E62">A81</font></i>: 
<font color="Maroon" title="c13">ma</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c21">1m</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green" title="E58">A77</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2_1"/>
<font color="Maroon" title="c19">cy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">1m</font>
 ;<br/><a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c20">mm</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">1m</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green" title="E57">A76</font></i></a>, <a class="txt" href="rearran1.html#E11:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green" title="E60">A79</font></i></a>, <a class="txt" href="rearran1.html#E2:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2"><i><font color="Green" title="E62">A81</font></i></a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c20">mm</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">1m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E4:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c19">cy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">1m</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"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">1m</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green" title="E57">A76</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="E5:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">cy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c20">mm</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="rearran1.html#E10:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green" title="E59">A78</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self" title="REARRAN1:th.22">Th22</a></i>;<br/><b>hence </b><a NAME="E6:36_1_1_1_2_3_1_2_1_2_1_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rearran1.html#E7:36_1_1_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_2_1"><i><font color="Green" title="E53">A71</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green" title="E57">A76</font></i></a>, <a class="txt" href="rearran1.html#E12:36_1_1_1_2_3_1_2_1_2_1_1"><i><font color="Green" title="E61">A80</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E14:36_1_1_1_2_3_1_2_1_2_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_2"/><i><font color="Green" title="E55">A82</font></i>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m1</font>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">m1</font> &amp; <font color="Maroon" title="c19">cy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 );<br/><a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_2"/><i><font color="Green" title="E56">A83</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>2</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2"><i><font color="Green" title="E50">A53</font></i></a>, <a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_2_1_2_1_2"><i><font color="Green" title="E55">A82</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c20">mm</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:36_1_1_1_2_3_1_2_1_2_1_2"/><i><font color="Green" title="E57">A84</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c20">mm</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>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c20">mm</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#E2:36_1_1_1_2_3_1_2_1_2_1_2"><i><font color="Green" title="E56">A83</font></i></a>);<br/></i><a NAME="E5:36_1_1_1_2_3_1_2_1_2_1_2"/><i><font color="Green" title="E58">A85</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">mm</font> &amp; <font color="Maroon" title="c20">mm</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> )
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_2"><i><font color="Green" title="E57">A84</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:36_1_1_1_2_3_1_2_1_2_1_2"/><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="c20">mm</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="c20">mm</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="c21">1m</font> = <font color="Maroon" title="c20">mm</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="E8:36_1_1_1_2_3_1_2_1_2_1_2"/><i><font color="Green" title="E59">A86</font></i>: 
<font color="Maroon" title="c20">mm</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> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_2"><i><font color="Green" title="E57">A84</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_2_1_1"/>
( <font color="Maroon" title="c20">mm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c20">mm</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="36_1_1_1_2_3_1_2_1_2_1_2_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_2_1_1_1"/>
<font color="Maroon" title="c20">mm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">cy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1
 <b>by </b><i><a class="txt" href="rearran1.html#E1:36"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_2"><i><font color="Green" title="E57">A84</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self" title="REARRAN1:th.22">Th22</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="rearran1.html#E14:36_1_1_1_2"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_2_1"><i><font color="Green" title="E53">A71</font></i></a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_2_3_1_2_1_2_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="rearran1.html#E1:36_1_1_1_2_3_1_2"><i><font color="Green" title="E48">A51</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/>
<font color="Maroon" title="c20">mm</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c20">mm</font> &amp; <font color="Maroon" title="c20">mm</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">mm</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c20">mm</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c20">mm</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_2_1_2_1_2"><i><font color="Green" title="E58">A85</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E60">A87</font></i>: 
( 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="c20">mm</font> &amp; <font color="Maroon" title="c21">1m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">mm</font> &amp; <font color="Maroon" title="c21">1m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c20">mm</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E4:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E61">A88</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">1m</font> &amp; <font color="Maroon" title="c21">1m</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> &amp; <font color="Maroon" title="c21">1m</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> &amp; <font color="Maroon" title="c21">1m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">m1</font> )
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_2"><i><font color="Green" title="E57">A84</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_2_1_2_1_2"><i><font color="Green" title="E58">A85</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c21">1m</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <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="E6:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
not <font color="Maroon" title="c19">cy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">1m</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_2"><i><font color="Green" title="E57">A84</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"><i><font color="Green" title="E60">A87</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"><i><font color="Green" title="E61">A88</font></i></a></i>;<br/><a NAME="E7:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c19">cy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">1m</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"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">1m</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_2"><i><font color="Green" title="E57">A84</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="E8:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">cy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">1m</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#E1:36"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"><i><font color="Green" title="E61">A88</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self" title="REARRAN1:th.22">Th22</a></i>;<br/><a NAME="E9:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">mm</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_2_1"><i><font color="Green" title="E53">A71</font></i></a>, <a class="txt" href="rearran1.html#E8:36_1_1_1_2_3_1_2_1_2_1_2"><i><font color="Green" title="E59">A86</font></i></a></i>;<br/><a NAME="E10:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c14">m1</font> <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="c14">m1</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:36_1_1_1_2_3_1_2_1_2_1_2"><i><font color="Green" title="E57">A84</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E11:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">m1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">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="E12:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>hence </b><a NAME="E13:36_1_1_1_2_3_1_2_1_2_1_2_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:36_1_1_1_2_3_1_2_1_2_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:36_1_1_1_2_3_1_2_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div><b>hence </b> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">bma</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c17">bm1</font></span>)</span>
<b>by </b><i><a class="txt" href="rearran1.html#E5:36_1_1_1_2_3_1_2"><i><font color="Green" title="E51">A54</font></i></a>, <a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></i>
<br/>.= 
<font color="Maroon" title="c13">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c17">bm1</font></span>)</span>
<b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="rearran1.html#E9:36_1_1_1_2"><i><font color="Green" title="E43">A12</font></i></a>, <a class="ref" href="rearran1.html#D1" target="_self" title="REARRAN1:def.1">Def1</a></i>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="rearran1.html#E3:36"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="rearran1.html#E19:36_1_1_1_2"><i><font color="Green" title="E47">A26</font></i></a>, <a class="txt" href="rearran1.html#E3:36_1_1_1_2_3_1_2"><i><font color="Green" title="E49">A52</font></i></a>, <a class="ref" href="rearran1.html#D1" target="_self" title="REARRAN1:def.1">Def1</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E21:36_1_1_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:36_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:36"/>
 <a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">B</font>, <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="ref" href="rfinseq.html#D1" title="RFINSEQ:def.1">RFINSEQ:def 1</a></i>;<br/>


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