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<b>let </b><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font> be  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:98_1_2"/><i><font color="Green" title="E53">A13</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>and </b><br/><a NAME="E2:98_1_2"/><i><font color="Green" title="E54">A14</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c3">g</font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E3:98_1_2"/><i><font color="Green" title="E55">A15</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">g</font>
 ;<br/>

<a NAME="E4:98_1_2"/>
( ( <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">g</font> &amp; <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> ) or ( <font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:98_1_2_1_1"/>
( <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">g</font> or <font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">f</font> )
 </span><b>by </b><i><a class="txt" href="newton.html#E3:98_1_2"><i><font color="Green" title="E55">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_1_2_1_1_1"><b>case </b></a><a NAME="E1:98_1_2_1_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">g</font>
 ;<br/><div class="add"><a NAME="E2:98_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E56">A16</font></i>: 
<span class="p1">(<span class="default"><a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">f</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c3">g</font>
 <b>by </b><i><a class="ref" href="newton.html#T94" target="_self" title="NEWTON:th.94">Th94</a></i>;<br/><a NAME="E3:98_1_2_1_1_1"/><i><font color="Green" title="E57">A17</font></i>: 
not <font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="ref" href="newton.html#D7" target="_self" title="NEWTON:def.7">Def7</a></i>;<br/><a NAME="E4:98_1_2_1_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">f</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c3">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="newton.html#E2:98_1_2_1_1_1"><i><font color="Green" title="E56">A16</font></i></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></i>;<br/><b>hence </b><a NAME="E5:98_1_2_1_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="newton.html#E1:98_1_2"><i><font color="Green" title="E53">A13</font></i></a>, <a class="txt" href="newton.html#E2:98_1_2"><i><font color="Green" title="E54">A14</font></i></a>, <a class="txt" href="newton.html#E3:98_1_2_1_1_1"><i><font color="Green" title="E57">A17</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_1_2_1_1_2"><b>case </b></a><a NAME="E1:98_1_2_1_1_2"/>
<font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">f</font>
 ;<br/><div class="add"><a NAME="E2:98_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E56">A18</font></i>: 
<span class="p1">(<span class="default"><a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">g</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="ref" href="newton.html#T94" target="_self" title="NEWTON:th.94">Th94</a></i>;<br/><a NAME="E3:98_1_2_1_1_2"/><i><font color="Green" title="E57">A19</font></i>: 
not <font color="Maroon" title="c3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c3">g</font>
 <b>by </b><i><a class="ref" href="newton.html#D7" target="_self" title="NEWTON:def.7">Def7</a></i>;<br/><a NAME="E4:98_1_2_1_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">g</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="newton.html#E2:98_1_2_1_1_2"><i><font color="Green" title="E56">A18</font></i></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></i>;<br/><b>hence </b><a NAME="E5:98_1_2_1_1_2"/>
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="newton.html#E1:98_1_2"><i><font color="Green" title="E53">A13</font></i></a>, <a class="txt" href="newton.html#E2:98_1_2"><i><font color="Green" title="E54">A14</font></i></a>, <a class="txt" href="newton.html#E3:98_1_2_1_1_2"><i><font color="Green" title="E57">A19</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:98_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>


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