<?xml version="1.0"?>
<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ( 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> <font color="Maroon">c<sub>1</sub></font> &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>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</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> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:22_1_1"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span><a href="mcart_1.html#K12">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:22_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span><a href="mcart_1.html#K12">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</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="E1:22_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:22_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i><a class="txt" href="heyting3.html#E2:22_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="grfunc_1.html#T6">GRFUNC_1:6</a></i>;<br/>
<a NAME="E5:22_1_1"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="txt" href="heyting3.html#E3:22_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="sysrel.html#T15">SYSREL:15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <b>by </b><i><a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E7:22_1_1"/>
(  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> or 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> <font color="Maroon">c<sub>1</sub></font> &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>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</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> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="heyting3.html#E2:22_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>

<b>assume </b><a NAME="E8:22_1_1"/><i><font color="Green">E15</font></i>: 
( 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> <font color="Maroon">c<sub>1</sub></font> &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>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</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> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_1_2"/>
( 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> <font color="Maroon">c<sub>1</sub></font> &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>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</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> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 </span><b>by </b><i><a class="txt" href="heyting3.html#E8:22_1_1"><i><font color="Green">E15</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_2_1"><b>suppose </b></a><a NAME="E1:22_1_1_2_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> <font color="Maroon">c<sub>1</sub></font> &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>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</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="E3:22_1_1_2_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E4:22_1_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i><a class="txt" href="heyting3.html#E3:22_1_1_2_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="zfmisc_1.html#T129">ZFMISC_1:129</a></i>;<br/><b>hence </b><a NAME="E5:22_1_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="heyting3.html#E2:22_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="heyting3.html#E8:22_1_1"><i><font color="Green">E15</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_2_2"><b>suppose </b></a><a NAME="E1:22_1_1_2_2"/><i><font color="Green">E16</font></i>: 
<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> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:22_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T129">ZFMISC_1:129</a></i>;<br/><b>hence </b><a NAME="E3:22_1_1_2_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="heyting3.html#E2:22_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="heyting3.html#E1:22_1_1_2_2"><i><font color="Green">E16</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
