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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1"/>
(  not <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1"><b>suppose </b></a><a NAME="E1:7_1_1"/><i><font color="Green">E7</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:7_1_1"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> holds <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:7_1_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="heyting3.html#E1:7_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="heyting3.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E5:7_1_1"/><i><font color="Green">E10</font></i>: 
not <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1_1_1"/><i><font color="Green">E11</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:7_1_1_1"/>
not <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:7_1_1_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="heyting3.html#E4:7_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="heyting3.html#E1:7_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:7_1_1_1"/><b>then </b><i><font color="Green">E12</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:7_1_1_1"/>
1 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T20">NAT_1:20</a></i>;<br/>
<b>hence </b><a NAME="E6:7_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="heyting3.html#E4:7_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:7_1_1"/>
not <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 is <a href="abian.html#V1">even</a>
 <b>by </b><i><a class="ref" href="abian.html#T9">ABIAN:9</a></i>;<br/><b>hence </b><a NAME="E7:7_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="heyting3.html#E2:7_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="heyting3.html#E5:7_1_1"><i><font color="Green">E10</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2"><b>suppose </b></a><a NAME="E1:7_1_2"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_2"/>
not for <font color="Olive">b<sub>1</sub></font> being <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> holds <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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