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<span class="kw">let </span><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font>, <font color="Maroon" title="c4">p3</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"> (  not <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_distinct</a>  or ( <font color="Maroon" title="c2">p1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c3">p2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 &amp; <font color="Maroon" title="c4">p3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 5 ) or ( <font color="Maroon" title="c2">p1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c3">p2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 5 &amp; <font color="Maroon" title="c4">p3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 ) or ( <font color="Maroon" title="c2">p1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 &amp; <font color="Maroon" title="c3">p2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c4">p3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 5 ) or ( <font color="Maroon" title="c2">p1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 &amp; <font color="Maroon" title="c3">p2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 5 &amp; <font color="Maroon" title="c4">p3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 ) or ( <font color="Maroon" title="c2">p1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 5 &amp; <font color="Maroon" title="c3">p2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c4">p3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 ) or ( <font color="Maroon" title="c2">p1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 5 &amp; <font color="Maroon" title="c3">p2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 &amp; <font color="Maroon" title="c4">p3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:76"/><span class="lab"><font color="Green" title="E59">A1</font></span>: 
<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_distinct</a> 
 <span class="kw">and </span><br/><a NAME="E2:76"/><span class="lab"><font color="Green" title="E60">A2</font></span>: 
( <font color="Maroon" title="c2">p1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 or <font color="Maroon" title="c3">p2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 3 or <font color="Maroon" title="c4">p3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 5 )
 <span class="kw">and </span><br/><a NAME="E3:76"/><span class="lab"><font color="Green" title="E61">A3</font></span>: 
( <font color="Maroon" title="c2">p1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 or <font color="Maroon" title="c3">p2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 5 or <font color="Maroon" title="c4">p3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 3 )
 <span class="kw">and </span><br/><a NAME="E4:76"/><span class="lab"><font color="Green" title="E62">A4</font></span>: 
( <font color="Maroon" title="c2">p1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 3 or <font color="Maroon" title="c3">p2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 or <font color="Maroon" title="c4">p3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 5 )
 <span class="kw">and </span><br/><a NAME="E5:76"/><span class="lab"><font color="Green" title="E63">A5</font></span>: 
( <font color="Maroon" title="c2">p1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 3 or <font color="Maroon" title="c3">p2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 5 or <font color="Maroon" title="c4">p3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 )
 <span class="kw">and </span><br/><a NAME="E6:76"/><span class="lab"><font color="Green" title="E64">A6</font></span>: 
( <font color="Maroon" title="c2">p1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 5 or <font color="Maroon" title="c3">p2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 or <font color="Maroon" title="c4">p3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 3 )
 <span class="kw">and </span><br/><a NAME="E7:76"/><span class="lab"><font color="Green" title="E65">A7</font></span>: 
( <font color="Maroon" title="c2">p1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 5 or <font color="Maroon" title="c3">p2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 3 or <font color="Maroon" title="c4">p3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_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="E1:76_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or ( <font color="Maroon" title="c2">p1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c3">p2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 3 ) or ( <font color="Maroon" title="c2">p1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c3">p2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 &amp; <font color="Maroon" title="c4">p3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 5 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:76"><span class="lab"><font color="Green" title="E60">A2</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1"><span class="kw">suppose </span></a><a NAME="E1:76_1_1"/>
<font color="Maroon" title="c2">p1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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"><a NAME="E2:76_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E3:76_1_1"/>
contradiction
 <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></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="76_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:76_1_2"/>
<font color="Maroon" title="c2">p1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <span class="kw">and </span><br/><a NAME="E2:76_1_2"/><span class="lab"><font color="Green" title="E66">A8</font></span>: 
<font color="Maroon" title="c3">p2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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"><a NAME="E3:76_1_2"/><span class="lab"><font color="Green" title="E67">A9</font></span>: 
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:76_1_2"><span class="lab"><font color="Green" title="E66">A8</font></span></a>, <a class="ref" href="number03.html#T65" onmouseover="rs('number03/T65')" onmouseout="rh()">NUMBER03:65</a></span>;<br/><a NAME="E4:76_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:76"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E4:76"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="ref" href="xprimet1.html#T3" onmouseover="rs('xprimet1/T3')" onmouseout="rh()">XPRIMET1:3</a>, <a class="ref" href="number03.html#T65" onmouseover="rs('number03/T65')" onmouseout="rh()">NUMBER03:65</a></span>;<br/><span class="kw">hence </span><a NAME="E5:76_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:76"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E6:76"><span class="lab"><font color="Green" title="E64">A6</font></span></a>, <a class="txt" href="#E3:76_1_2"><span class="lab"><font color="Green" title="E67">A9</font></span></a>, <a class="ref" href="xprimet1.html#T3" onmouseover="rs('xprimet1/T3')" onmouseout="rh()">XPRIMET1:3</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="76_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:76_1_3"/><span class="lab"><font color="Green" title="E66">A10</font></span>: 
<font color="Maroon" title="c2">p1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <span class="kw">and </span><br/><a NAME="E2:76_1_3"/><span class="lab"><font color="Green" title="E67">A11</font></span>: 
<font color="Maroon" title="c3">p2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3
 <span class="kw">and </span><br/><a NAME="E3:76_1_3"/><span class="lab"><font color="Green" title="E68">A12</font></span>: 
<font color="Maroon" title="c4">p3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 5
 ; <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"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_3_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="E1:76_1_3_1"/>
( <font color="Maroon" title="c4">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c4">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:76_1_3"><span class="lab"><font color="Green" title="E68">A12</font></span></a>, <a class="ref" href="xprimet1.html#T3" onmouseover="rs('xprimet1/T3')" onmouseout="rh()">XPRIMET1:3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:76_1_3_1_1"/><span class="lab"><font color="Green" title="E69">A13</font></span>: 
<font color="Maroon" title="c4">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><a NAME="E2:76_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:76"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E7:76"><span class="lab"><font color="Green" title="E65">A7</font></span></a>, <a class="txt" href="#E2:76_1_3"><span class="lab"><font color="Green" title="E67">A11</font></span></a>, <a class="ref" href="xprimet1.html#T3" onmouseover="rs('xprimet1/T3')" onmouseout="rh()">XPRIMET1:3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:76_1_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:76"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E5:76"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="txt" href="#E1:76_1_3_1_1"><span class="lab"><font color="Green" title="E69">A13</font></span></a>, <a class="ref" href="xprimet1.html#T3" onmouseover="rs('xprimet1/T3')" onmouseout="rh()">XPRIMET1:3</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="76_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:76_1_3_1_2"/>
<font color="Maroon" title="c4">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><a NAME="E2:76_1_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:76"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E3:76"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="txt" href="#E1:76_1_3"><span class="lab"><font color="Green" title="E66">A10</font></span></a>, <a class="ref" href="xprimet1.html#T3" onmouseover="rs('xprimet1/T3')" onmouseout="rh()">XPRIMET1:3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:76_1_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:76_1_3"><span class="lab"><font color="Green" title="E67">A11</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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