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<div class="add">

<b>consider </b><font color="Maroon" title="c2">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E2:42_1_1"/><i><font color="Green" title="E26">A6</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">k</font>
 <b>by </b><i><a class="ref" href="afinsq_2.html#T2" target="_self" title="AFINSQ_2:th.2">AE221f</a></i>;<br/>
<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>for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> $1 holds <br/> ex <font color="Olive" title="b2">p</font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> &amp; ( for <font color="Olive" title="b3">l</font>, <font color="Olive" title="b4">m</font>, <font color="Olive" title="b5">k1</font>, <font color="Olive" title="b6">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">m</font> &amp; <font color="Olive" title="b4">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b5">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">l</font> &amp; <font color="Olive" title="b6">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">m</font> holds <br/><font color="Olive" title="b5">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b6">k2</font> ) );<br/>
<a NAME="E3:42_1_1"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  ex <font color="Olive" title="b1">p</font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; ( for <font color="Olive" title="b2">l</font>, <font color="Olive" title="b3">m</font>, <font color="Olive" title="b4">k1</font>, <font color="Olive" title="b5">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">m</font> &amp; <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">l</font> &amp; <font color="Olive" title="b5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">m</font> holds <br/><font color="Olive" title="b4">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">k2</font> ) ) )</span><br/>

<b>assume </b><a NAME="E1:42_1_1_1"/><i><font color="Green" title="E28">A3</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; ( for <font color="Olive" title="b2">l</font>, <font color="Olive" title="b3">m</font>, <font color="Olive" title="b4">k1</font>, <font color="Olive" title="b5">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">m</font> &amp; <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">l</font> &amp; <font color="Olive" title="b5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">m</font> holds <br/><font color="Olive" title="b4">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">k2</font> ) )</span><br/>

<b>take </b>
 <a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; ( for <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k1</font>, <font color="Olive" title="b4">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> &amp; <font color="Olive" title="b3">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b4">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b3">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">k2</font> ) )</span><br/>

<b>thus </b><a NAME="E2:42_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="#E1:42_1_1_1"><i><font color="Green" title="E28">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k1</font>, <font color="Olive" title="b4">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> &amp; <font color="Olive" title="b3">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b4">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b3">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">k2</font></span><br/>

<b>thus </b><a NAME="E3:42_1_1_1"/>
for <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k1</font>, <font color="Olive" title="b4">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> &amp; <font color="Olive" title="b3">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b4">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b3">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">k2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:42_1_1"/><i><font color="Green" title="E28">A4</font></i>: 
for <font color="Olive" title="b1">k</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">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:42_1_1_2"/><i><font color="Green" title="E29">A5</font></i>: 
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">k</font> holds <br/> ex <font color="Olive" title="b2">p</font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> &amp; ( for <font color="Olive" title="b3">l</font>, <font color="Olive" title="b4">m</font>, <font color="Olive" title="b5">k1</font>, <font color="Olive" title="b6">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">m</font> &amp; <font color="Olive" title="b4">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b5">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">l</font> &amp; <font color="Olive" title="b6">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">m</font> holds <br/><font color="Olive" title="b5">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b6">k2</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<b>let </b><font color="Maroon" title="c4">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies  ex <font color="Olive" title="b1">p</font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font> &amp; ( for <font color="Olive" title="b2">l</font>, <font color="Olive" title="b3">m</font>, <font color="Olive" title="b4">k1</font>, <font color="Olive" title="b5">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">m</font> &amp; <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">l</font> &amp; <font color="Olive" title="b5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">m</font> holds <br/><font color="Olive" title="b4">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">k2</font> ) ) )</span><br/>

<b>assume </b><a NAME="E2:42_1_1_2"/><i><font color="Green" title="E30">A6</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font> &amp; ( for <font color="Olive" title="b2">l</font>, <font color="Olive" title="b3">m</font>, <font color="Olive" title="b4">k1</font>, <font color="Olive" title="b5">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">m</font> &amp; <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">l</font> &amp; <font color="Olive" title="b5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">m</font> holds <br/><font color="Olive" title="b4">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">k2</font> ) )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1_2_1"/>
not <font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font> &amp; ( for <font color="Olive" title="b2">l</font>, <font color="Olive" title="b3">m</font>, <font color="Olive" title="b4">k1</font>, <font color="Olive" title="b5">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">m</font> &amp; <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">l</font> &amp; <font color="Olive" title="b5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">m</font> holds <br/><font color="Olive" title="b4">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">k2</font> ) )</span><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="E3:42_1_1_2_1"/><i><font color="Green" title="E31">A7</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">k</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">n</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="ref" href="afinsq_2.html#T1" target="_self" title="AFINSQ_2:th.1">AB470</a>, <a class="txt" href="#E2:42_1_1_2"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="#E3:42_1_1_2_1"><i><font color="Green" title="E31">A7</font></i></a></i>;<br/><a NAME="E5:42_1_1_2_1"/>
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="#E2:42_1_1_2"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="#E3:42_1_1_2_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/><a NAME="E6:42_1_1_2_1"/><b>then </b><i><font color="Green" title="E32">A19</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</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="E7:42_1_1_2_1"/>
not <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="#E3:42_1_1_2_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/><a NAME="E8:42_1_1_2_1"/><b>then </b><i><font color="Green" title="E33">A8</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="#E6:42_1_1_2_1"><i><font color="Green" title="E32">A19</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>set </b><font color="Maroon" title="c7">Y</font> = <font color="Maroon" title="c4">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">k</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><a NAME="E9:42_1_1_2_1"/><i><font color="Green" title="E34">A10</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">k</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">k</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">k</font> )</span><br/>

<b>assume </b><a NAME="E1:42_1_1_2_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">k</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">k</font></span><br/>

<a NAME="E2:42_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E35">A12n</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; not <font color="Maroon" title="c8">x</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="c3">k</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">m</font> = <font color="Maroon" title="c8">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="ref" href="afinsq_2.html#T1" target="_self" title="AFINSQ_2:th.1">AB470</a>, <a class="txt" href="#E2:42_1_1_2"><i><font color="Green" title="E30">A6</font></i></a></i>;<br/>
<a NAME="E4:42_1_1_2_1_1"/>
( <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c9">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">k</font> )
 
<b>by </b><i><a class="txt" href="#E2:42_1_1_2_1_1"><i><font color="Green" title="E35">A12n</font></i></a>, <a class="txt" href="#E2:42_1_1_2"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:42_1_1_2_1_1"/><b>then </b>
( <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c9">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">k</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="E6:42_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E7:42_1_1_2_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c8">p</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E11:42_1_1_2_1"/><i><font color="Green" title="E35">A13</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">k</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; ( for <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k1</font>, <font color="Olive" title="b4">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">p</font> &amp; <font color="Olive" title="b3">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b4">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b3">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">k2</font> ) )
 <b>by </b><i><a class="txt" href="#E1:42_1_1_2"><i><font color="Green" title="E29">A5</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">k</font> = <font color="Maroon" title="c3">k</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="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">q</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E14:42_1_1_2_1"/><i><font color="Green" title="E36">A14</font></i>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c9">k</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
 ;<br/><a NAME="E15:42_1_1_2_1"/><i><font color="Green" title="E37">A15</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">k</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="#E3:42_1_1_2_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="#E8:42_1_1_2_1"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><i><font color="Green" title="E38">A16</font></i>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">q</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c9">k</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E14:42_1_1_2_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="ref" href="afinsq_1.html#T29" title="AFINSQ_1:th.29">AFINSQ_1:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">k</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">k</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="#E11:42_1_1_2_1"><i><font color="Green" title="E35">A13</font></i></a>, <a class="ref" href="afinsq_1.html#T36" title="AFINSQ_1:th.36">AFINSQ_1:36</a></i>
<br/>.= 
<font color="Maroon" title="c4">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">k</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></i>
<br/>.= 
<font color="Maroon" title="c4">X</font>
<b>by </b><i><a class="txt" href="#E15:42_1_1_2_1"><i><font color="Green" title="E37">A15</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
;<br/><a NAME="E17:42_1_1_2_1"/>
for <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k1</font>, <font color="Olive" title="b4">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c10">q</font> &amp; <font color="Olive" title="b3">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b4">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b3">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">k2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">l</font>, <font color="Maroon" title="c12">m</font>, <font color="Maroon" title="c13">k1</font>, <font color="Maroon" title="c14">k2</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">m</font> &amp; <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c10">q</font> &amp; <font color="Maroon" title="c13">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">l</font> &amp; <font color="Maroon" title="c14">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">m</font> implies <font color="Maroon" title="c13">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">k2</font> )</span><br/>

<b>assume </b><a NAME="E1:42_1_1_2_1_3"/><i><font color="Green" title="E39">A17</font></i>: 
( <font color="Maroon" title="c11">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">m</font> &amp; <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c10">q</font> &amp; <font color="Maroon" title="c13">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">l</font> &amp; <font color="Maroon" title="c14">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">m</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">k2</font></span><br/>

<a NAME="E2:42_1_1_2_1_3"/><b>then </b>
<font color="Maroon" title="c12">m</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="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c10">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="E3:42_1_1_2_1_3"/><b>then </b><i><font color="Green" title="E40">A38</font></i>: 
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c10">q</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="E4:42_1_1_2_1_3"/><i><font color="Green" title="E41">A38b</font></i>: 
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c10">q</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<b>by </b><i><a class="txt" href="#E3:42_1_1_2_1_3"><i><font color="Green" title="E40">A38</font></i></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></i>;<br/>
<a NAME="E5:42_1_1_2_1_3"/><i><font color="Green" title="E42">A18</font></i>: 
<font color="Maroon" title="c11">l</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="c8">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:42_1_1_2_1_3_1"/>
<font color="Maroon" title="c11">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p3"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c9">k</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<b>by </b><i><a class="txt" href="#E3:42_1_1_2_1_3"><i><font color="Green" title="E40">A38</font></i></a>, <a class="txt" href="#E14:42_1_1_2_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="txt" href="#E1:42_1_1_2_1_3"><i><font color="Green" title="E39">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E2:42_1_1_2_1_3_1"/><b>then </b>
<font color="Maroon" title="c11">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c9">k</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T20" title="AFINSQ_1:th.20">AFINSQ_1:20</a></i>;<br/>
<a NAME="E3:42_1_1_2_1_3_1"/><b>then </b>
<font color="Maroon" title="c11">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></i>;<br/>
<b>hence </b><a NAME="E4:42_1_1_2_1_3_1"/>
<font color="Maroon" title="c11">l</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="c8">p</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E43">A20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1_2_1_3_2"/><i><font color="Green" title="E44">A21</font></i>: 
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c10">q</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">k2</font></span><br/><a NAME="E2:42_1_1_2_1_3_2"/>
<font color="Maroon" title="c13">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">l</font>
 <b>by </b><i><a class="txt" href="#E14:42_1_1_2_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="txt" href="#E1:42_1_1_2_1_3"><i><font color="Green" title="E39">A17</font></i></a>, <a class="txt" href="#E5:42_1_1_2_1_3"><i><font color="Green" title="E42">A18</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E3:42_1_1_2_1_3_2"/><b>then </b><i><font color="Green" title="E45">A22n</font></i>: 
<font color="Maroon" title="c13">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">k</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E11:42_1_1_2_1"><i><font color="Green" title="E35">A13</font></i></a>, <a class="txt" href="#E5:42_1_1_2_1_3"><i><font color="Green" title="E42">A18</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><font color="Maroon" title="c10">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c9">k</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p4"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c9">k</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E14:42_1_1_2_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="txt" href="#E1:42_1_1_2_1_3_2"><i><font color="Green" title="E44">A21</font></i></a>, <a class="ref" href="afinsq_1.html#T20" title="AFINSQ_1:th.20">AFINSQ_1:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c9">k</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c9">k</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">p</font></span>)</span>
<b>by </b><i><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></i>
<br/>.= 
<font color="Maroon" title="c9">k</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#T40" title="AFINSQ_1:th.40">AFINSQ_1:40</a></i>
;<br/><b>hence </b><a NAME="E5:42_1_1_2_1_3_2"/>
<font color="Maroon" title="c13">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">k2</font>
 <b>by </b><i><a class="txt" href="#E1:42_1_1_2_1_3"><i><font color="Green" title="E39">A17</font></i></a>, <a class="txt" href="#E9:42_1_1_2_1"><i><font color="Green" title="E34">A10</font></i></a>, <a class="txt" href="#E3:42_1_1_2_1_3_2"><i><font color="Green" title="E45">A22n</font></i></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_1_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1_2_1_3_3"/>
<font color="Maroon" title="c12">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c10">q</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">k2</font></span><br/><a NAME="E2:42_1_1_2_1_3_3"/><b>then </b>
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p3"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c9">k</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <b>by </b><i><a class="txt" href="#E4:42_1_1_2_1_3"><i><font color="Green" title="E41">A38b</font></i></a>, <a class="txt" href="#E14:42_1_1_2_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E3:42_1_1_2_1_3_3"/><b>then </b>
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c9">k</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <b>by </b><i><a class="ref" href="afinsq_1.html#T20" title="AFINSQ_1:th.20">AFINSQ_1:20</a></i>;<br/><a NAME="E4:42_1_1_2_1_3_3"/><b>then </b>
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <b>by </b><i><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></i>;<br/><a NAME="E5:42_1_1_2_1_3_3"/><b>then </b><i><font color="Green" title="E44">A23</font></i>: 
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">p</font>
 <b>by </b><i><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></i>;<br/><a NAME="E6:42_1_1_2_1_3_3"/><b>then </b>
<font color="Maroon" title="c12">m</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="c8">p</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/><a NAME="E7:42_1_1_2_1_3_3"/><b>then </b><i><font color="Green" title="E45">A24</font></i>: 
<font color="Maroon" title="c14">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">m</font>
 <b>by </b><i><a class="txt" href="#E14:42_1_1_2_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="txt" href="#E1:42_1_1_2_1_3"><i><font color="Green" title="E39">A17</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E8:42_1_1_2_1_3_3"/>
<font color="Maroon" title="c13">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">l</font>
 <b>by </b><i><a class="txt" href="#E14:42_1_1_2_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="txt" href="#E1:42_1_1_2_1_3"><i><font color="Green" title="E39">A17</font></i></a>, <a class="txt" href="#E5:42_1_1_2_1_3"><i><font color="Green" title="E42">A18</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><b>hence </b><a NAME="E9:42_1_1_2_1_3_3"/>
<font color="Maroon" title="c13">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">k2</font>
 <b>by </b><i><a class="txt" href="#E11:42_1_1_2_1"><i><font color="Green" title="E35">A13</font></i></a>, <a class="txt" href="#E1:42_1_1_2_1_3"><i><font color="Green" title="E39">A17</font></i></a>, <a class="txt" href="#E5:42_1_1_2_1_3_3"><i><font color="Green" title="E44">A23</font></i></a>, <a class="txt" href="#E7:42_1_1_2_1_3_3"><i><font color="Green" title="E45">A24</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:42_1_1_2_1_3"/>
<font color="Maroon" title="c13">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">k2</font>
 <b>by </b><i><a class="txt" href="#E6:42_1_1_2_1_3"><i><font color="Green" title="E43">A20</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E18:42_1_1_2_1"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font> &amp; ( for <font color="Olive" title="b2">l</font>, <font color="Olive" title="b3">m</font>, <font color="Olive" title="b4">k1</font>, <font color="Olive" title="b5">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">m</font> &amp; <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">l</font> &amp; <font color="Olive" title="b5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">m</font> holds <br/><font color="Olive" title="b4">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">k2</font> ) )
 <b>by </b><i><a class="txt" href="#E16:42_1_1_2_1"><i><font color="Green" title="E38">A16</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:42_1_1_2"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font> &amp; ( for <font color="Olive" title="b2">l</font>, <font color="Olive" title="b3">m</font>, <font color="Olive" title="b4">k1</font>, <font color="Olive" title="b5">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">m</font> &amp; <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">l</font> &amp; <font color="Olive" title="b5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">m</font> holds <br/><font color="Olive" title="b4">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">k2</font> ) )
 <b>by </b><i><a class="txt" href="#E1:42_1_1_2"><i><font color="Green" title="E29">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:42_1_1"/>
for <font color="Olive" title="b1">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k2</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E3:42_1_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="#E4:42_1_1"><i><font color="Green" title="E28">A4</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:42_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  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b2">l</font>, <font color="Olive" title="b3">m</font>, <font color="Olive" title="b4">k1</font>, <font color="Olive" title="b5">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">m</font> &amp; <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive" title="b4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">l</font> &amp; <font color="Olive" title="b5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">m</font> holds <br/><font color="Olive" title="b4">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">k2</font> ) )
 <b>by </b><i><a class="txt" href="#E2:42_1_1"><i><font color="Green" title="E26">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
