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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>thus </b><a NAME="E1:37"/>
 <a href="heyting3.html#K2" target="_self">PFArt</a> 1,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">1,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default">2,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K7">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> 1,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1"/>
( ex <font color="Olive">b<sub>1</sub></font> being <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> 1 &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> ) or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="heyting3.html#E1:37_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="heyting3.html#D2" target="_self">Def2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> 1 &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:37_1_1_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> 1 &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><a NAME="E4:37_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="txt" href="heyting3.html#E3:37_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="heyting3.html#E3"><i><font color="Green">Lemma3</font></i></a>, <a class="ref" href="cqc_the1.html#T3">CQC_THE1:3</a></i>;<br/><b>hence </b><a NAME="E5:37_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">1,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default">2,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">1,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default">2,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:37"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">1,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default">2,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K7">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">2,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> )
 </span><b>by </b><i><a class="txt" href="heyting3.html#E2:37"><i><font color="Green">E24</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1"><b>suppose </b></a><a NAME="E1:37_2_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span>
 ;<br/><div class="add"><a NAME="E2:37_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> 1
 ;<br/><b>hence </b><a NAME="E3:37_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> 1,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="heyting3.html#E1:37_2_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="heyting3.html#D2" target="_self">Def2</a>, <a class="txt" href="heyting3.html#E5"><i><font color="Green">Lemma4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_2"><b>suppose </b></a><a NAME="E1:37_2_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">2,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span>
 ;<br/><div class="add"><a NAME="E2:37_2_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span>
 ;<br/><b>hence </b><a NAME="E3:37_2_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> 1,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="heyting3.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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