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<b>let </b><font color="Maroon" title="c1">i</font> be    <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> ;<br/>

<b>thus </b><a NAME="E1:14"/>
( <a href="scmpds_1.html#K2" title="SCMPDS_1:func.2">SCMPDS-OK</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_1.html#K1" title="SCMPDS_1:func.1">SCMPDS-Instr</a>  implies  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" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1"/><i><font color="Green" title="E12">A1</font></i>: 
<a href="scmpds_1.html#K2" title="SCMPDS_1:func.2">SCMPDS-OK</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_1.html#K1" title="SCMPDS_1:func.1">SCMPDS-Instr</a> 
 ;<br/>

<b>assume </b><a NAME="E2:14_1"/><i><font color="Green" title="E13">A2</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds not <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2
 ;<br/>

<a NAME="E3:14_1"/><i><font color="Green" title="E14">A3</font></i>: 
for <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>   holds not <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c2">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> <b> such that </b><a NAME="E1:14_1_1"/><i><font color="Green" title="E15">A4</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 ;<br/>

<b>thus </b><a NAME="E2:14_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_1.html#E2:14_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="scmpds_1.html#E1:14_1_1"><i><font color="Green" title="E15">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_2"/>
( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or  ex <font color="Olive" title="b1">j</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 <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="scmpds_1.html#E3:14_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="scmpds_1.html#T14" target="_self" title="SCMPDS_1:th.14">Th14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1"><b>suppose </b></a><a NAME="E1:14_1_2_1"/>
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:14_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_1.html#E1:14_1"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="scmpds_1.html#D4" target="_self" title="SCMPDS_1:def.4">Def4</a>, <a class="ref" href="scmpds_1.html#T17" target="_self" title="SCMPDS_1:th.17">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_2"><b>suppose </b></a><a NAME="E1:14_1_2_2"/>
 ex <font color="Olive" title="b1">j</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 <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c2">j</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> <b> such that </b><br/><a NAME="E3:14_1_2_2"/><i><font color="Green" title="E15">A5</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><b>thus </b><a NAME="E4:14_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_1.html#E1:14_1"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="scmpds_1.html#E3:14_1_2_2"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="scmpds_1.html#D4" target="_self" title="SCMPDS_1:def.4">Def4</a>, <a class="ref" href="scmpds_1.html#T17" target="_self" title="SCMPDS_1:th.17">Th17</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><a NAME="E2:14"/>
(  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" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2 implies <a href="scmpds_1.html#K2" title="SCMPDS_1:func.2">SCMPDS-OK</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_1.html#K1" title="SCMPDS_1:func.1">SCMPDS-Instr</a>  )
 <b>by </b><i><a class="ref" href="scmpds_1.html#D4" target="_self" title="SCMPDS_1:def.4">Def4</a></i>;<br/>


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