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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="real_1.html#NM1">Real</a>] <b>means </b> ex <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> &amp; <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  );<br/>
<b>consider </b><font color="Maroon">X</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E2:3_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] )
 <b>from </b><i><a class="ref" href="real_1.html#S1">REAL_1:sch 1</a>();<br/></i>
<a NAME="E3:3_1_1"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
;<br/>
<a NAME="E4:3_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
0 <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:3_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<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="E3:3_1_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i/>;<br/><a NAME="E4:3_1_1_1"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 ;<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="E6:3_1_1_1"/>
( <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i>, <a class="ref" href="axioms.html#T28">AXIOMS:28</a></i>;<br/><b>hence </b><a NAME="E7:3_1_1_1"/>
<font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<a NAME="E6:3_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:3_1_1"/><b>then </b>
 ex <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> &amp; <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">m</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:3_1_1"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/>


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