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<span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="abian.html#V1" title="ABIAN:attr.1">odd</a>  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 iff  <a href="newton05.html#K2" title="NEWTON05:func.2">parity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="newton05.html#K2" title="NEWTON05:func.2">parity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> 2</span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E1:132"/>
(  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies  <a href="newton05.html#K2" title="NEWTON05:func.2">parity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="newton05.html#K2" title="NEWTON05:func.2">parity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> 2</span>)</span> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="newton05.html#K2" title="NEWTON05:func.2">parity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="newton05.html#K2" title="NEWTON05:func.2">parity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> 2</span>)</span> implies  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="132_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:132_1"/>
 <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="newton05.html#K2" title="NEWTON05:func.2">parity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="newton05.html#K2" title="NEWTON05:func.2">parity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> 2</span>)</span></span><br/>

<a NAME="E2:132_1"/><span class="kw">then </span>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton05.html#T94" target="_self" onmouseover="rs('newton05/T94')" onmouseout="rh()">SPA</a></span>;<br/>
<a NAME="E3:132_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> 2</span>)</span> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E4:132_1"/>
 <a href="newton05.html#K2" title="NEWTON05:func.2">parity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="newton05.html#K2" title="NEWTON05:func.2">parity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton05.html#T57" target="_self" onmouseover="rs('newton05/T57')" onmouseout="rh()">EVP</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>

<span class="kw">assume </span><a NAME="E2:132"/>
 <a href="newton05.html#K2" title="NEWTON05:func.2">parity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="newton05.html#K2" title="NEWTON05:func.2">parity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> 2</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2</span><br/>

<a NAME="E3:132"/><span class="kw">then </span>
( <font color="Maroon" title="c2">a</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> 2 is  <a href="abian.html#V1" title="ABIAN:attr.1">odd</a>  iff <font color="Maroon" title="c3">b</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> 2 is  <a href="abian.html#V1" title="ABIAN:attr.1">odd</a>  )
 
;<br/>
<a NAME="E4:132"/><span class="kw">then </span>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#D1" onmouseover="rs('nat_2/D1')" onmouseout="rh()">NAT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:132"/>
 <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton05.html#T94" target="_self" onmouseover="rs('newton05/T94')" onmouseout="rh()">SPA</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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