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<b>assume </b><a NAME="E1:12_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>4</sub></font>()
 ;<br/>

<a NAME="E2:12_1"/><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:12_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F<sub>4</sub></font>() <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:12_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E6:12_1"/><i><font color="Green">E38</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:12_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:12_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:12_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:12_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><a NAME="E7:12_1_1"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>,<font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">F<sub>4</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>,<font color="Maroon">F<sub>4</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E8:12_1_1"/>
contradiction
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:12_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


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