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<span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</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> <font color="Maroon" title="c2">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c3">b</font> iff  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<a NAME="E1:57"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
(  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default">2 <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</font></span>)</span> &amp;  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default">2 <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">b</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton05.html#D3" target="_self" onmouseover="rs('newton05/D3')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E2:57"/>
(  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c2">a</font> implies  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c2">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c3">b</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="57_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:57_1"/>
 <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c2">a</font>
 ; <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> <font color="Maroon" title="c2">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c3">b</font></span><br/>

<a NAME="E2:57_1"/><span class="kw">then </span>
2 <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="pepin.html#T66" onmouseover="rs('pepin/T66')" onmouseout="rh()">PEPIN:66</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:57_1"/>
 <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c2">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="newton.html#T89" onmouseover="rs('newton/T89')" onmouseout="rh()">NEWTON:89</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">hence </span><a NAME="E3:57"/>
(  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c2">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c3">b</font> iff  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="newton05.html#K3" title="NEWTON05:func.3">Parity</a> <font color="Maroon" title="c2">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T7" onmouseover="rs('nat_d/T7')" onmouseout="rh()">NAT_D:7</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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