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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>] <span class="kw">means </span> ex <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">k</font> is  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  );<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">X</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E2:3_1_1"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="subset_1.html#S3" title="SUBSET_1:sch.3">SUBSET_1:sch 3</a>();<br/></span>
<div><span class="lab"><font color="Green" title="E3">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )</span><br/><span class="kw">assume </span><a NAME="E1:3_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">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="E3:3_1_1_1"/><span class="lab"><font color="Green" title="E4">A3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">and </span><br/><a NAME="E4:3_1_1_1"/><span class="lab"><font color="Green" title="E5">A4</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">k</font> is  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_1_1"><span class="lab"><font color="Green" title="E2">A1</font></span></a></span>;<br/><a NAME="E5:3_1_1_1"/><span class="lab"><font color="Green" title="E6">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">k</font> = <font color="Maroon" title="c5">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="E7:3_1_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">k</font> is    <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="txt" href="#E4:3_1_1_1"><span class="lab"><font color="Green" title="E5">A4</font></span></a>, <a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E8:3_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 is    <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>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 is    <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="axioms.html#T2" title="AXIOMS:th.2">AXIOMS:2</a></span>;<br/><span class="kw">hence </span><a NAME="E9:3_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_1_1"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E3:3_1_1_1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E5:3_1_1_1"><span class="lab"><font color="Green" title="E6">A5</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>
<a NAME="E4:3_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
;<br/>
<a NAME="E5:3_1_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_1_1"><span class="lab"><font color="Green" title="E2">A1</font></span></a></span>;<br/>
<a NAME="E6:3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_1"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="ref" href="nat_1.html#T1" target="_self" title="NAT_1:th.1">Th1</a></span>;<br/>
<a NAME="E7:3_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">m</font> is  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_1_1"><span class="lab"><font color="Green" title="E2">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:3_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">k</font> is  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> 
 ; <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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