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<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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[$1] );<br/>
<b>assume </b><a NAME="E1:1_1"/>
 ex <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 <br/>( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] )
 ;<br/>

<a NAME="E2:1_1"/><b>then </b><i><font color="Green" title="E3">A3</font></i>: 
 ex <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>]
 
;<br/>
<b>reconsider </b><font color="Maroon" title="c1">n'</font> = <font color="Maroon">F<sub>1</sub></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/>
<a NAME="E4:1_1"/><i><font color="Green" title="E4">A4</font></i>: 
for <font color="Olive" title="b1">l</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">l</font>] holds <br/><font color="Olive" title="b1">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n'</font>
 
;<br/>
<b>consider </b><font color="Maroon" title="c2">l</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:1_1"/><i><font color="Green" title="E5">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">l</font>]
 <b>and </b><br/><a NAME="E7:1_1"/><i><font color="Green" title="E6">A6</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> <font color="Maroon" title="c2">l</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="triang_1.html#E4:1_1"><i><font color="Green" title="E4">A4</font></i></a>, <a class="txt" href="triang_1.html#E2:1_1"><i><font color="Green" title="E3">A3</font></i></a>);<br/></i>
<a NAME="E8:1_1"/><i><font color="Green" title="E7">A7</font></i>: 
<font color="Maroon" title="c2">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="txt" href="triang_1.html#E1:1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="txt" href="triang_1.html#E6:1_1"><i><font color="Green" title="E5">A5</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="E9:1_1"/><b>then </b><i><font color="Green" title="E8">A8</font></i>: 
<font color="Maroon" title="c2">l</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">F<sub>1</sub></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="E10:1_1"/><i><font color="Green" title="E9">A9</font></i>: 
<font color="Maroon" title="c2">l</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/>
<a NAME="E11:1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<b>by </b><i><a class="txt" href="triang_1.html#E7:1_1"><i><font color="Green" title="E6">A6</font></i></a>, <a class="txt" href="triang_1.html#E9:1_1"><i><font color="Green" title="E8">A8</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<b>hence </b><a NAME="E12:1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="triang_1.html#E2:1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="triang_1.html#E6:1_1"><i><font color="Green" title="E5">A5</font></i></a>, <a class="txt" href="triang_1.html#E8:1_1"><i><font color="Green" title="E7">A7</font></i></a>, <a class="txt" href="triang_1.html#E10:1_1"><i><font color="Green" title="E9">A9</font></i></a></i>;<br/>


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