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<b>let </b><font color="Maroon" title="c1">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c2">R</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>;<br/><b>let </b><font color="Maroon" title="c3">p</font> be    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of <font color="Maroon" title="c2">R</font>;<br/>





<b>assume </b><a NAME="E1:3"/><i><font color="Green" title="E4">A1</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1"/>
( <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> or <font color="Maroon" title="c1">k</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="c3">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1"><b>suppose </b></a><a NAME="E1:3_1_1"/>
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:3_1_1"/>
<font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">k</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of <font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_2"><b>suppose </b></a><a NAME="E1:3_1_2"/>
<font color="Maroon" title="c1">k</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="c3">p</font>
 ;<br/><div class="add"><a NAME="E2:3_1_2"/><b>then </b><i><font color="Green" title="E5">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="rewrite2.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">i</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="E1:3_1_2_1"/><i><font color="Green" title="E6">A3</font></i>: 
( <font color="Maroon" title="c4">i</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"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">k</font></span>)</span> &amp; <font color="Maroon" title="c4">i</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"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">k</font></span>)</span> )
 ;<br/><a NAME="E2:3_1_2_1"/><i><font color="Green" title="E7">A4</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/><a NAME="E3:3_1_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</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="c4">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">k</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="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="rewrite2.html#E1:3_1_2_1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E4:3_1_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">k</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="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="txt" href="rewrite2.html#E1:3_1_2_1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="rewrite2.html#E2:3_1_2_1"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="rewrite1.html#D2" title="REWRITE1:def.2">REWRITE1:def 2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:3_1_2"/>
<font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">k</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of <font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="txt" href="rewrite2.html#E2:3_1_2"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="rewrite1.html#D2" title="REWRITE1:def.2">REWRITE1:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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