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<span class="kw">given </span><font color="Maroon" title="c17">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><a NAME="E2:98"/><span class="lab"><font color="Green" title="E81">A1</font></span>: 
<font color="Maroon" title="c17">n</font> <a href="number07.html#R6" title="NUMBER07:pred.6">satisfies_Sierpinski_problem_89</a> 
 <span class="kw">and </span><br/><a NAME="E3:98"/>
<font color="Maroon" title="c17">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="number07.html#R6" title="NUMBER07:pred.6">satisfies_Sierpinski_problem_89</a> 
 <span class="kw">and </span><br/><a NAME="E4:98"/>
<font color="Maroon" title="c17">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2 <a href="number07.html#R6" title="NUMBER07:pred.6">satisfies_Sierpinski_problem_89</a> 
 <span class="kw">and </span><br/><a NAME="E5:98"/><span class="lab"><font color="Green" title="E82">A2</font></span>: 
<font color="Maroon" title="c17">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 3 <a href="number07.html#R6" title="NUMBER07:pred.6">satisfies_Sierpinski_problem_89</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"> contradiction</span><br/>

<a NAME="E6:98"/><span class="lab"><font color="Green" title="E83">A3</font></span>: 
<font color="Maroon" title="c17">n</font> <a href="number07.html#R5" title="NUMBER07:pred.5">is_a_product_of_two_different_primes</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E81">A1</font></span></a></span>;<br/>
<a NAME="E7:98"/><span class="lab"><font color="Green" title="E84">A4</font></span>: 
<font color="Maroon" title="c17">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="number07.html#R5" title="NUMBER07:pred.5">is_a_product_of_two_different_primes</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E81">A1</font></span></a></span>;<br/>
<a NAME="E8:98"/><span class="lab"><font color="Green" title="E85">A5</font></span>: 
<font color="Maroon" title="c17">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2 <a href="number07.html#R5" title="NUMBER07:pred.5">is_a_product_of_two_different_primes</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E81">A1</font></span></a></span>;<br/>
<a NAME="E9:98"/><span class="lab"><font color="Green" title="E86">A6</font></span>: 
<font color="Maroon" title="c17">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 3 <a href="number07.html#R5" title="NUMBER07:pred.5">is_a_product_of_two_different_primes</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98"><span class="lab"><font color="Green" title="E82">A2</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_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="E1:98_1"/>
( 4 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c17">n</font> or 4 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c17">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 or 4 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c17">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2 or 4 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c17">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 3 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="number06.html#T19" onmouseover="rs('number06/T19')" onmouseout="rh()">NUMBER06:19</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_1_1"><span class="kw">suppose </span></a><a NAME="E1:98_1_1"/><span class="lab"><font color="Green" title="E87">A7</font></span>: 
4 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c17">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"> contradiction</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c18">p</font>, <font color="Maroon" title="c19">q</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a><span class="kw"> such that </span><br/><a NAME="E3:98_1_1"/><span class="lab"><font color="Green" title="E88">A8</font></span>: 
<font color="Maroon" title="c18">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">q</font>
 <span class="kw">and </span><br/><a NAME="E4:98_1_1"/><span class="lab"><font color="Green" title="E89">A9</font></span>: 
<font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:98"><span class="lab"><font color="Green" title="E83">A3</font></span></a></span>;<br/><a NAME="E5:98_1_1"/><span class="lab"><font color="Green" title="E90">A10</font></span>: 
( <font color="Maroon" title="c18">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c18">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">q</font> &amp; <font color="Maroon" title="c19">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c18">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">q</font> )
 ;<br/><a NAME="E6:98_1_1"/>
4 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c18">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98_1_1"><span class="lab"><font color="Green" title="E89">A9</font></span></a>, <a class="txt" href="#E1:98_1_1"><span class="lab"><font color="Green" title="E87">A7</font></span></a></span>;<br/><a NAME="E7:98_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gr_cy_3.html#T1" onmouseover="rs('gr_cy_3/T1')" onmouseout="rh()">GR_CY_3:1</a>, <a class="ref" href="xprimes0.html#T4" onmouseover="rs('xprimes0/T4')" onmouseout="rh()">XPRIMES0:4</a></span>;<br/><a NAME="E8:98_1_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c18">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c18">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) &amp; ( <font color="Maroon" title="c19">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c19">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98_1_1"><span class="lab"><font color="Green" title="E90">A10</font></span></a>, <a class="ref" href="number05.html#T21" onmouseover="rs('number05/T21')" onmouseout="rh()">NUMBER05:21</a></span>;<br/><span class="kw">hence </span><a NAME="E9:98_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98_1_1"><span class="lab"><font color="Green" title="E88">A8</font></span></a>, <a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</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="98_1_2"><span class="kw">suppose </span></a><a NAME="E1:98_1_2"/><span class="lab"><font color="Green" title="E87">A11</font></span>: 
4 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c17">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ; <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/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c18">p</font>, <font color="Maroon" title="c19">q</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a><span class="kw"> such that </span><br/><a NAME="E3:98_1_2"/><span class="lab"><font color="Green" title="E88">A12</font></span>: 
<font color="Maroon" title="c18">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">q</font>
 <span class="kw">and </span><br/><a NAME="E4:98_1_2"/><span class="lab"><font color="Green" title="E89">A13</font></span>: 
<font color="Maroon" title="c17">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:98"><span class="lab"><font color="Green" title="E84">A4</font></span></a></span>;<br/><a NAME="E5:98_1_2"/><span class="lab"><font color="Green" title="E90">A14</font></span>: 
( <font color="Maroon" title="c18">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c18">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">q</font> &amp; <font color="Maroon" title="c19">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c18">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">q</font> )
 ;<br/><a NAME="E6:98_1_2"/>
4 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c18">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98_1_2"><span class="lab"><font color="Green" title="E89">A13</font></span></a>, <a class="txt" href="#E1:98_1_2"><span class="lab"><font color="Green" title="E87">A11</font></span></a></span>;<br/><a NAME="E7:98_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gr_cy_3.html#T1" onmouseover="rs('gr_cy_3/T1')" onmouseout="rh()">GR_CY_3:1</a>, <a class="ref" href="xprimes0.html#T4" onmouseover="rs('xprimes0/T4')" onmouseout="rh()">XPRIMES0:4</a></span>;<br/><a NAME="E8:98_1_2"/><span class="kw">then </span>
( ( <font color="Maroon" title="c18">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c18">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) &amp; ( <font color="Maroon" title="c19">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c19">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98_1_2"><span class="lab"><font color="Green" title="E90">A14</font></span></a>, <a class="ref" href="number05.html#T21" onmouseover="rs('number05/T21')" onmouseout="rh()">NUMBER05:21</a></span>;<br/><span class="kw">hence </span><a NAME="E9:98_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98_1_2"><span class="lab"><font color="Green" title="E88">A12</font></span></a>, <a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</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="98_1_3"><span class="kw">suppose </span></a><a NAME="E1:98_1_3"/><span class="lab"><font color="Green" title="E87">A15</font></span>: 
4 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c17">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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"> contradiction</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c18">p</font>, <font color="Maroon" title="c19">q</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a><span class="kw"> such that </span><br/><a NAME="E3:98_1_3"/><span class="lab"><font color="Green" title="E88">A16</font></span>: 
<font color="Maroon" title="c18">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">q</font>
 <span class="kw">and </span><br/><a NAME="E4:98_1_3"/><span class="lab"><font color="Green" title="E89">A17</font></span>: 
<font color="Maroon" title="c17">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:98"><span class="lab"><font color="Green" title="E85">A5</font></span></a></span>;<br/><a NAME="E5:98_1_3"/><span class="lab"><font color="Green" title="E90">A18</font></span>: 
( <font color="Maroon" title="c18">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c18">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">q</font> &amp; <font color="Maroon" title="c19">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c18">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">q</font> )
 ;<br/><a NAME="E6:98_1_3"/>
4 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c18">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98_1_3"><span class="lab"><font color="Green" title="E89">A17</font></span></a>, <a class="txt" href="#E1:98_1_3"><span class="lab"><font color="Green" title="E87">A15</font></span></a></span>;<br/><a NAME="E7:98_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c18">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gr_cy_3.html#T1" onmouseover="rs('gr_cy_3/T1')" onmouseout="rh()">GR_CY_3:1</a>, <a class="ref" href="xprimes0.html#T4" onmouseover="rs('xprimes0/T4')" onmouseout="rh()">XPRIMES0:4</a></span>;<br/><a NAME="E8:98_1_3"/><span class="kw">then </span>
( ( <font color="Maroon" title="c18">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c18">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) &amp; ( <font color="Maroon" title="c19">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c19">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98_1_3"><span class="lab"><font color="Green" title="E90">A18</font></span></a>, <a class="ref" href="number05.html#T21" onmouseover="rs('number05/T21')" onmouseout="rh()">NUMBER05:21</a></span>;<br/><span class="kw">hence </span><a NAME="E9:98_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98_1_3"><span class="lab"><font color="Green" title="E88">A16</font></span></a>, <a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</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="98_1_4"><span class="kw">suppose </span></a><a NAME="E1:98_1_4"/><span class="lab"><font color="Green" title="E87">A19</font></span>: 
4 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c17">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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"> contradiction</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c18">p</font>, <font color="Maroon" title="c19">q</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a><span class="kw"> such that </span><br/><a NAME="E3:98_1_4"/><span class="lab"><font color="Green" title="E88">A20</font></span>: 
<font color="Maroon" title="c18">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">q</font>
 <span class="kw">and </span><br/><a NAME="E4:98_1_4"/><span class="lab"><font color="Green" title="E89">A21</font></span>: 
<font color="Maroon" title="c17">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:98"><span class="lab"><font color="Green" title="E86">A6</font></span></a></span>;<br/><a NAME="E5:98_1_4"/><span class="lab"><font color="Green" title="E90">A22</font></span>: 
( <font color="Maroon" title="c18">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c18">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">q</font> &amp; <font color="Maroon" title="c19">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c18">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">q</font> )
 ;<br/><a NAME="E6:98_1_4"/>
4 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c18">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98_1_4"><span class="lab"><font color="Green" title="E89">A21</font></span></a>, <a class="txt" href="#E1:98_1_4"><span class="lab"><font color="Green" title="E87">A19</font></span></a></span>;<br/><a NAME="E7:98_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c18">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gr_cy_3.html#T1" onmouseover="rs('gr_cy_3/T1')" onmouseout="rh()">GR_CY_3:1</a>, <a class="ref" href="xprimes0.html#T4" onmouseover="rs('xprimes0/T4')" onmouseout="rh()">XPRIMES0:4</a></span>;<br/><a NAME="E8:98_1_4"/><span class="kw">then </span>
( ( <font color="Maroon" title="c18">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c18">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) &amp; ( <font color="Maroon" title="c19">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c19">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98_1_4"><span class="lab"><font color="Green" title="E90">A22</font></span></a>, <a class="ref" href="number05.html#T21" onmouseover="rs('number05/T21')" onmouseout="rh()">NUMBER05:21</a></span>;<br/><span class="kw">hence </span><a NAME="E9:98_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98_1_4"><span class="lab"><font color="Green" title="E88">A20</font></span></a>, <a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</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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