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<div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>;<br/>
<a NAME="E1:3_1_1"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a> st <br/>(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">q</font> &amp; ( for <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">q</font> holds <br/><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">k</font>) ) )
 
<b>from </b><i><a class="ref" href="afinsq_1.html#S2" title="AFINSQ_1:sch.2">AFINSQ_1:sch 2</a>();<br/></i>
<b>then consider </b><font color="Maroon" title="c3">p</font> being   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a><b> such that </b><br/><a NAME="E3:3_1_1"/><i><font color="Green" title="E2">A1</font></i>: 
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">q</font> &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">q</font> holds <br/><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) ) )
 ;<br/>
<a NAME="E4:3_1_1"/><i><font color="Green" title="E3">A2</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">q</font> holds <br/><font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<a NAME="E1:3_1_1_1"/><i><font color="Green" title="E4">A3</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:3_1_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">q</font>
 ;<br/>

<a NAME="E3:3_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">q</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T98" title="CARD_1:th.98">CARD_1:98</a></i>;<br/>
<b>hence </b><a NAME="E4:3_1_1_1"/>
<font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="prgcor_2.html#E1:3_1_1_1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="prgcor_2.html#E3:3_1_1"><i><font color="Green" title="E2">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:3_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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="E1:3_1_1_2"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">p</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:3_1_1_2"/><i><font color="Green" title="E4">A4</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:3_1_1_2"/><i><font color="Green" title="E5">A5</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E3:3_1_1_2"><i><font color="Green" title="E4">A4</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">nx</font> = <font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="prgcor_2.html#E3:3_1_1_2"><i><font color="Green" title="E4">A4</font></i></a></i>;<br/>
<a NAME="E6:3_1_1_2"/><i><font color="Green" title="E6">A6</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">nx</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E3:3_1_1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="prgcor_2.html#E4:3_1_1_2"><i><font color="Green" title="E5">A5</font></i></a></i>;<br/>
<a NAME="E7:3_1_1_2"/>
<font color="Maroon" title="c6">nx</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">q</font>
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E3:3_1_1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="prgcor_2.html#E4:3_1_1_2"><i><font color="Green" title="E5">A5</font></i></a>, <a class="ref" href="card_1.html#T98" title="CARD_1:th.98">CARD_1:98</a></i>;<br/>
<a NAME="E8:3_1_1_2"/><b>then </b><i><font color="Green" title="E7">A7</font></i>: 
<font color="Maroon" title="c6">nx</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">q</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="E9:3_1_1_2"/>
0 <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="c6">nx</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E10:3_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">nx</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E8:3_1_1_2"><i><font color="Green" title="E7">A7</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E11:3_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">nx</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> <font color="Maroon" title="c2">q</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E12:3_1_1_2"/><b>then </b><i><font color="Green" title="E8">A8</font></i>: 
<font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">nx</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">q</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E13:3_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E14:3_1_1_2"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font>
 <b>by </b><i><a class="txt" href="prgcor_2.html#E3:3_1_1_2"><i><font color="Green" title="E4">A4</font></i></a>, <a class="txt" href="prgcor_2.html#E6:3_1_1_2"><i><font color="Green" title="E6">A6</font></i></a>, <a class="txt" href="prgcor_2.html#E12:3_1_1_2"><i><font color="Green" title="E8">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:3_1_1"/><b>then </b>
<font color="Maroon" title="c3">p</font> is   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c1">D</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D8" title="ORDINAL1:def.8">ORDINAL1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E7:3_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c1">D</font> st <br/>(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">q</font> &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">q</font> holds <br/><font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> ) )
 <b>by </b><i><a class="txt" href="prgcor_2.html#E3:3_1_1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="prgcor_2.html#E4:3_1_1"><i><font color="Green" title="E3">A2</font></i></a></i>;<br/>


</div>
