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<b>let </b><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</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/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1"/>
( <font color="Maroon" title="c1">m</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> or  not <font color="Maroon" title="c1">m</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1"><b>suppose </b></a><a NAME="E1:18_1_1"/>
<font color="Maroon" title="c1">m</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c3">m</font> = <font color="Maroon" title="c1">m</font> as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <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="E3:18_1_1"/>
( ( <font color="Maroon" title="c1">m</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> iff <font color="Maroon" title="c2">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> ) iff <font color="Maroon" title="c1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1_1_1"/>
( <font color="Maroon" title="c2">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> or  not <font color="Maroon" title="c2">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:18_1_1_1_1_1"/>
<font color="Maroon" title="c2">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c4">n</font> = <font color="Maroon" title="c2">n</font> as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <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/><a NAME="E3:18_1_1_1_1_1"/>
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><b>hence </b><a NAME="E4:18_1_1_1_1_1"/>
( ( <font color="Maroon" title="c1">m</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> iff <font color="Maroon" title="c2">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> ) iff <font color="Maroon" title="c1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:18_1_1_1_1_2"/>
not <font color="Maroon" title="c2">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c4">n</font> = <font color="Maroon" title="c2">n</font> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <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/><a NAME="E3:18_1_1_1_1_2"/>
not <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><b>hence </b><a NAME="E4:18_1_1_1_1_2"/>
( ( <font color="Maroon" title="c1">m</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> iff <font color="Maroon" title="c2">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> ) iff <font color="Maroon" title="c1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2"><b>suppose </b></a><a NAME="E1:18_1_2"/>
not <font color="Maroon" title="c1">m</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c3">m</font> = <font color="Maroon" title="c1">m</font> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <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="E3:18_1_2"/>
( ( <font color="Maroon" title="c1">m</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> iff <font color="Maroon" title="c2">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> ) iff <font color="Maroon" title="c1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_2_1_1"/>
( <font color="Maroon" title="c2">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> or  not <font color="Maroon" title="c2">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:18_1_2_1_1_1"/>
<font color="Maroon" title="c2">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c4">n</font> = <font color="Maroon" title="c2">n</font> as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <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/><a NAME="E3:18_1_2_1_1_1"/>
not <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><b>hence </b><a NAME="E4:18_1_2_1_1_1"/>
( ( <font color="Maroon" title="c1">m</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> iff <font color="Maroon" title="c2">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> ) iff <font color="Maroon" title="c1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:18_1_2_1_1_2"/>
not <font color="Maroon" title="c2">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c4">n</font> = <font color="Maroon" title="c2">n</font> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <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/><a NAME="E3:18_1_2_1_1_2"/>
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><b>hence </b><a NAME="E4:18_1_2_1_1_2"/>
( ( <font color="Maroon" title="c1">m</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> iff <font color="Maroon" title="c2">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> ) iff <font color="Maroon" title="c1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

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