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<span class="kw">let </span><font color="Maroon" title="c1">a</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">  ex <font color="Olive" title="b1">n</font> being   non  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <font color="Olive" title="b1">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:49_1"/>
(  not <font color="Maroon" title="c1">a</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  or ( <font color="Maroon" title="c1">a</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 ) or ( <font color="Maroon" title="c1">a</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1"><span class="kw">suppose </span></a><a NAME="E1:49_1_1"/>
 not <font color="Maroon" title="c1">a</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</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">  ex <font color="Olive" title="b1">n</font> being   non  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <font color="Olive" title="b1">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">n</font> = <font color="Maroon" title="c1">a</font> as   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>   non  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><span class="kw">take </span>
<font color="Maroon" title="c2">n</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"> <font color="Maroon" title="c2">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span><br/><a NAME="E3:49_1_1"/>
<font color="Maroon" title="c1">a</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number02.html#T18" target="_self" onmouseover="rs('number02/T18')" onmouseout="rh()">Th18</a></span>;<br/><span class="kw">hence </span><a NAME="E4:49_1_1"/>
<font color="Maroon" title="c2">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:49_1_2"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
<font color="Maroon" title="c1">a</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 <span class="kw">and </span><br/><a NAME="E2:49_1_2"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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">  ex <font color="Olive" title="b1">n</font> being   non  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <font color="Olive" title="b1">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c2">a</font> = <font color="Maroon" title="c1">a</font> as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_1_2"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E2:49_1_2"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="pepin.html#T17" onmouseover="rs('pepin/T17')" onmouseout="rh()">PEPIN:17</a></span>;<br/><a NAME="E4:49_1_2"/>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/><a NAME="E5:49_1_2"/><span class="kw">then </span>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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="txt" href="#E2:49_1_2"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">n</font> = 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">a</font> as   non  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T17" onmouseover="rs('pepin/T17')" onmouseout="rh()">PEPIN:17</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c3">n</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"> <font color="Maroon" title="c3">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span><br/><a NAME="E7:49_1_2"/><span class="lab"><font color="Green" title="E40">A3</font></span>: 
2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 1
 ;<br/><a NAME="E8:49_1_2"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">a</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 ;<br/><a NAME="E9:49_1_2"/>
<font color="Maroon" title="c2">a</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number02.html#T18" target="_self" onmouseover="rs('number02/T18')" onmouseout="rh()">Th18</a></span>;<br/><span class="kw">hence </span><a NAME="E10:49_1_2"/>
<font color="Maroon" title="c3">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49_1_2"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E8:49_1_2"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="ref" href="pepin.html#T4" onmouseover="rs('pepin/T4')" onmouseout="rh()">PEPIN:4</a>, <a class="ref" href="nat_5.html#T3" onmouseover="rs('nat_5/T3')" onmouseout="rh()">NAT_5:3</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:49_1_3"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
<font color="Maroon" title="c1">a</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 <span class="kw">and </span><br/><a NAME="E2:49_1_3"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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">  ex <font color="Olive" title="b1">n</font> being   non  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <font color="Olive" title="b1">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span><br/><div class="add"><a NAME="E3:49_1_3"/>
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_1_3"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:49_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A7</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49_1_3"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E5:49_1_3"/><span class="lab"><font color="Green" title="E41">A8</font></span>: 
341 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 11 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 31
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">n</font> = 341 as   non  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_4.html#T27" onmouseover="rs('nat_4/T27')" onmouseout="rh()">NAT_4:27</a>, <a class="ref" href="numeral2.html#T24" onmouseover="rs('numeral2/T24')" onmouseout="rh()">NUMERAL2:24</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c2">n</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"> <font color="Maroon" title="c2">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span><br/><span class="kw">thus </span><a NAME="E7:49_1_3"/>
<font color="Maroon" title="c2">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49_1_3"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E5:49_1_3"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="ref" href="number02.html#T20" target="_self" onmouseover="rs('number02/T20')" onmouseout="rh()">Th20</a>, <a class="ref" href="number02.html#T21" target="_self" onmouseover="rs('number02/T21')" onmouseout="rh()">Th21</a>, <a class="ref" href="pepin.html#T4" onmouseover="rs('pepin/T4')" onmouseout="rh()">PEPIN:4</a>, <a class="ref" href="int_2.html#T30" onmouseover="rs('int_2/T30')" onmouseout="rh()">INT_2:30</a>, <a class="ref" href="nat_4.html#T27" onmouseover="rs('nat_4/T27')" onmouseout="rh()">NAT_4:27</a>, <a class="ref" href="numeral2.html#T24" onmouseover="rs('numeral2/T24')" onmouseout="rh()">NUMERAL2:24</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></div><span class="kw">end;</span></div>

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