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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#NV3" title="ABIAN:NV.3">odd</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>  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#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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/>
<a NAME="E1:24_1_2"/><i><font color="Green" title="E14">A4</font></i>: 
for <font color="Olive" title="b1">X1</font>, <font color="Olive" title="b2">X2</font> being   <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>  st ( for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X1</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">x</font>] ) ) &amp; ( for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">X2</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">x</font>] ) ) holds <br/><font color="Olive" title="b1">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">X2</font>
 
<b>from </b><i><a class="ref" href="lmod_7.html#S1" title="LMOD_7:sch.1">LMOD_7:sch 1</a>();<br/></i>
<b>let </b><font color="Maroon" title="c3">X1</font>, <font color="Maroon" title="c4">X2</font> be    <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>;<br/>

<b>assume </b><a NAME="E2:24_1_2"/><i><font color="Green" title="E15">A5</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">X1</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) ) &amp; ( 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="c4">X2</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) ) )
 ;<br/>

<b>thus </b><a NAME="E3:24_1_2"/>
<font color="Maroon" title="c3">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X2</font>
 <b>by </b><i><a class="txt" href="heyting3.html#E1:24_1_2"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="heyting3.html#E2:24_1_2"><i><font color="Green" title="E15">A5</font></i></a></i>;<br/>


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