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<span class="kw">assume </span><a NAME="E1:97"/>
10 <a href="number08.html#R2" title="NUMBER08:pred.2">satisfies_Sierpinski_problem_86</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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_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="E1:97_1"/><span class="kw">then </span>
( ( 10 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp;  ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; 10 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">y</font> ) ) or ( 10 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp;  ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; 10 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">y</font> ) ) )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#T52" target="_self" onmouseover="rs('number08/T52')" onmouseout="rh()">Th52</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:97_1_1"/>
10 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 <span class="kw">and </span><br/><a NAME="E2:97_1_1"/><span class="lab"><font color="Green" title="E74">A1</font></span>: 
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; 10 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">y</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">thus </span><a NAME="E3:97_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97_1_1"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="ref" href="xprimes1.html#T11" onmouseover="rs('xprimes1/T11')" onmouseout="rh()">XPRIMES1:11</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="97_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:97_1_2"/>
10 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 <span class="kw">and </span><br/><a NAME="E2:97_1_2"/><span class="lab"><font color="Green" title="E74">A2</font></span>: 
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; 10 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">y</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="c2">x</font>, <font color="Maroon" title="c3">y</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a><span class="kw"> such that </span><br/><a NAME="E4:97_1_2"/><span class="lab"><font color="Green" title="E75">A3</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">and </span><br/><a NAME="E5:97_1_2"/><span class="lab"><font color="Green" title="E76">A4</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 9
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97_1_2"><span class="lab"><font color="Green" title="E74">A2</font></span></a></span>;<br/><a NAME="E6:97_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">y</font>
 ;<br/><a NAME="E7:97_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 9 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:97_1_2"><span class="lab"><font color="Green" title="E76">A4</font></span></a>, <a class="ref" href="number08.html#T9" target="_self" onmouseover="rs('number08/T9')" onmouseout="rh()">Th9</a></span>;<br/><span class="kw">hence </span><a NAME="E8:97_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97_1_2"><span class="lab"><font color="Green" title="E75">A3</font></span></a>, <a class="txt" href="#E5:97_1_2"><span class="lab"><font color="Green" title="E76">A4</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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