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<span class="kw">thus </span><a NAME="E1:102"/><span class="lab"><font color="Green" title="E81">A1</font></span>: 
36 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 3</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 3
 ; <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"> 36 <a href="number07.html#R4" title="NUMBER07:pred.4">has_exactly_two_different_prime_divisors</a> </span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c1">P2</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="number07.html#D4" target="_self" onmouseover="rs('number07/D4')" onmouseout="rh()">NUMBER07:def 4</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">q</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">P2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">q</font> &amp; <font color="Maroon" title="c1">P2</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 36 &amp; <font color="Olive" title="b1">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 36 &amp; (  for <font color="Olive" title="b2">r</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>  st <font color="Olive" title="b2">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">P2</font> &amp; <font color="Olive" title="b2">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">q</font> holds <br/> not <font color="Olive" title="b2">r</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 36 ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c2">P3</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="c1">P2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">P3</font> &amp; <font color="Maroon" title="c1">P2</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 36 &amp; <font color="Maroon" title="c2">P3</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 36 &amp; (  for <font color="Olive" title="b1">r</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">P2</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">P3</font> holds <br/> not <font color="Olive" title="b1">r</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 36 ) )</span><br/>

<span class="kw">thus </span><a NAME="E2:102"/>
<font color="Maroon" title="c1">P2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">P3</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="c1">P2</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 36 &amp; <font color="Maroon" title="c2">P3</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 36 &amp; (  for <font color="Olive" title="b1">r</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">P2</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">P3</font> holds <br/> not <font color="Olive" title="b1">r</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 36 ) )</span><br/>

<a NAME="E3:102"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 3</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 3</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 3</span>)</span>
 
;<br/>
<span class="kw">hence </span><a NAME="E4:102"/>
<font color="Maroon" title="c1">P2</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 36
 ; <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">P3</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 36 &amp; (  for <font color="Olive" title="b1">r</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">P2</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">P3</font> holds <br/> not <font color="Olive" title="b1">r</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 36 ) )</span><br/>

<span class="kw">thus </span><a NAME="E5:102"/>
<font color="Maroon" title="c2">P3</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 36
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102"><span class="lab"><font color="Green" title="E81">A1</font></span></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">  for <font color="Olive" title="b1">r</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">P2</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">P3</font> holds <br/> not <font color="Olive" title="b1">r</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 36</span><br/>

<span class="kw">let </span><font color="Maroon" title="c17">r</font> be   <a href="int_2.html#NM1" title="INT_2:NM.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"> ( <font color="Maroon" title="c17">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">P2</font> &amp; <font color="Maroon" title="c17">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">P3</font> implies  not <font color="Maroon" title="c17">r</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 36 )</span><br/>

<span class="kw">assume </span><a NAME="E6:102"/><span class="lab"><font color="Green" title="E82">A2</font></span>: 
( <font color="Maroon" title="c17">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">P2</font> &amp; <font color="Maroon" title="c17">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">P3</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">  not <font color="Maroon" title="c17">r</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 36</span><br/>

<span class="kw">assume </span><a NAME="E7:102"/>
<font color="Maroon" title="c17">r</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 36
 ; <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"> contradiction</span><br/>

<a NAME="E8:102"/><span class="kw">then </span>
( <font color="Maroon" title="c17">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c17">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c17">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c17">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or <font color="Maroon" title="c17">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6 or <font color="Maroon" title="c17">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 9 or <font color="Maroon" title="c17">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 12 or <font color="Maroon" title="c17">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 18 or <font color="Maroon" title="c17">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 36 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number07.html#T15" target="_self" onmouseover="rs('number07/T15')" onmouseout="rh()">Th15</a></span>;<br/>
<a NAME="E9:102"/><span class="kw">then </span>
( <font color="Maroon" title="c17">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c17">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</a>, <a class="ref" href="xprimes0.html#T4" onmouseover="rs('xprimes0/T4')" onmouseout="rh()">XPRIMES0:4</a>, <a class="ref" href="xprimes0.html#T6" onmouseover="rs('xprimes0/T6')" onmouseout="rh()">XPRIMES0:6</a>, <a class="ref" href="xprimes0.html#T9" onmouseover="rs('xprimes0/T9')" onmouseout="rh()">XPRIMES0:9</a>, <a class="ref" href="xprimes0.html#T12" onmouseover="rs('xprimes0/T12')" onmouseout="rh()">XPRIMES0:12</a>, <a class="ref" href="xprimes0.html#T18" onmouseover="rs('xprimes0/T18')" onmouseout="rh()">XPRIMES0:18</a>, <a class="ref" href="xprimes0.html#T36" onmouseover="rs('xprimes0/T36')" onmouseout="rh()">XPRIMES0:36</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:102"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:102"><span class="lab"><font color="Green" title="E82">A2</font></span></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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