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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">m</font> being <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">m</font>,<font color="Maroon" title="c2">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 );<br/>
<b>consider </b><font color="Maroon" title="c3">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:24_1_1"/><i><font color="Green" title="E12">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">k</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:24_1_1"/><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">k</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:24_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<b>hence </b><a NAME="E2:24_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">k</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 <b>by </b><i><a class="txt" href="heyting3.html#E2:24_1_1"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">X'</font> = <font color="Maroon" title="c3">X</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="ref" href="grfunc_1.html#T6" title="GRFUNC_1:th.6">GRFUNC_1:6</a></i>;<br/>
<a NAME="E5:24_1_1"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">X'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">X'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">k</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="txt" href="heyting3.html#E3:24_1_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="sysrel.html#T15" title="SYSREL:th.15">SYSREL:15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">X</font> = <font color="Maroon" title="c3">X</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">k</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <b>by </b><i><a class="ref" href="partfun1.html#D5" title="PARTFUN1:def.5">PARTFUN1:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">X</font>
;<br/>

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E7:24_1_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> implies  ex <font color="Olive" title="b1">m</font> being <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">m</font>,<font color="Maroon" title="c2">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> ) )
 <b>by </b><i><a class="txt" href="heyting3.html#E2:24_1_1"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/>

<b>given </b><font color="Maroon" title="c7">m</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><a NAME="E9:24_1_1"/><i><font color="Green" title="E14">A3</font></i>: 
( <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c2">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> )
 ;<br/>

<a NAME="E10:24_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">k</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="txt" href="heyting3.html#E9:24_1_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T129" title="ZFMISC_1:th.129">ZFMISC_1:129</a></i>;<br/>
<b>hence </b><a NAME="E11:24_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="txt" href="heyting3.html#E2:24_1_1"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="heyting3.html#E9:24_1_1"><i><font color="Green" title="E14">A3</font></i></a></i>;<br/>


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