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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<a NAME="E1:75_1"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon">F<sub>4</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:75_1"/>
<font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">F<sub>4</sub></font>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:75_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>;<br/>
<a NAME="E4:75_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A7</font></span>: 
 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/>
<span class="kw">consider </span><font color="Maroon" title="c1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:75_1"/><span class="lab"><font color="Green" title="E37">A8</font></span>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">m</font>] &amp; (  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="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" target="_self" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E4:75_1"><span class="lab"><font color="Green" title="E36">A7</font></span></a>);<br/></span>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:75_1_1"/>
<font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:75_1_1"/><span class="lab"><font color="Green" title="E38">A9</font></span>: 
<font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T6" target="_self" title="NAT_1:th.6">Th6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c3">k</font> = <font color="Maroon" title="c2">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E5:75_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75_1_1"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="ref" href="nat_1.html#T13" target="_self" title="NAT_1:th.13">Th13</a></span>;<br/><a NAME="E6:75_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A10</font></span>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:75_1"><span class="lab"><font color="Green" title="E37">A8</font></span></a></span>;<br/><a NAME="E7:75_1_1"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>,<font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>,<font color="Maroon">F<sub>4</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon">F<sub>4</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E4:75"><span class="lab"><font color="Green" title="E33">A4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:75_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E6:75_1"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E3:75_1_1"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="txt" href="#E6:75_1_1"><span class="lab"><font color="Green" title="E39">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:75_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E3:75"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E6:75_1"><span class="lab"><font color="Green" title="E37">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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