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<b>let </b><font color="Maroon">i</font> be   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">j</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:3_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> iff <font color="Maroon">j</font> is <a href="abian.html#V1">even</a> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1_1"/>
( <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> or  not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1"><b>suppose </b></a><a NAME="E1:3_1_1_1"/>
<font color="Maroon">i</font> is <a href="abian.html#V1">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font>, <font color="Maroon">j'</font> = <font color="Maroon">j</font> as  <a href="abian.html#V1">even</a> <a href="int_1.html#NM1">Integer</a> <b>by </b><i/>;<br/><a NAME="E3:3_1_1_1"/>
<font color="Maroon">i'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j'</font> is <a href="abian.html#V1">even</a>
 ;<br/><b>hence </b><a NAME="E4:3_1_1_1"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font> is <a href="abian.html#V1">even</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_2"><b>suppose </b></a><a NAME="E1:3_1_1_2"/>
not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font>, <font color="Maroon">j'</font> = <font color="Maroon">j</font> as  <a href="abian.html#V1">odd</a> <a href="int_1.html#NM1">Integer</a> <b>by </b><i/>;<br/><a NAME="E3:3_1_1_2"/>
<font color="Maroon">i'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j'</font> is <a href="abian.html#V1">even</a>
 ;<br/><b>hence </b><a NAME="E4:3_1_1_2"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font> is <a href="abian.html#V1">even</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:3"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font> is <a href="abian.html#V1">even</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_2"/>
( ( <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">j</font> is <a href="abian.html#V1">even</a> ) or ( <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; not <font color="Maroon">j</font> is <a href="abian.html#V1">even</a> ) or ( not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">j</font> is <a href="abian.html#V1">even</a> ) or ( not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; not <font color="Maroon">j</font> is <a href="abian.html#V1">even</a> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1"><b>suppose </b></a><a NAME="E1:3_2_1"/>
( <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">j</font> is <a href="abian.html#V1">even</a> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:3_2_1"/>
( <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> iff <font color="Maroon">j</font> is <a href="abian.html#V1">even</a> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_2"><b>suppose </b></a><a NAME="E1:3_2_2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; not <font color="Maroon">j</font> is <a href="abian.html#V1">even</a> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as  <a href="abian.html#V1">even</a> <a href="int_1.html#NM1">Integer</a> ;<br/><b>reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j</font> as  <a href="abian.html#V1">odd</a> <a href="int_1.html#NM1">Integer</a> <b>by </b><i/>;<br/><a NAME="E4:3_2_2"/>
not <font color="Maroon">i'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j'</font> is <a href="abian.html#V1">even</a>
 ;<br/><b>hence </b><a NAME="E5:3_2_2"/>
( <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> iff <font color="Maroon">j</font> is <a href="abian.html#V1">even</a> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_3"><b>suppose </b></a><a NAME="E1:3_2_3"/><i><font color="Green">E37</font></i>: 
( not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">j</font> is <a href="abian.html#V1">even</a> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as  <a href="abian.html#V1">odd</a> <a href="int_1.html#NM1">Integer</a> ;<br/><b>reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j</font> as  <a href="abian.html#V1">even</a> <a href="int_1.html#NM1">Integer</a> <b>by </b><i/>;<br/><a NAME="E4:3_2_3"/>
not <font color="Maroon">i'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j'</font> is <a href="abian.html#V1">even</a>
 ;<br/><b>hence </b><a NAME="E5:3_2_3"/>
( <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> iff <font color="Maroon">j</font> is <a href="abian.html#V1">even</a> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_4"><b>suppose </b></a><a NAME="E1:3_2_4"/>
( not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> &amp; not <font color="Maroon">j</font> is <a href="abian.html#V1">even</a> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:3_2_4"/>
( <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> iff <font color="Maroon">j</font> is <a href="abian.html#V1">even</a> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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