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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>] <span class="kw">means </span> ex <font color="Olive" title="b1">z</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Olive" title="b1">z</font> );<br/>
<a NAME="E1:56_1_1"/>
 ex <font color="Olive" title="b1">B</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>for <font color="Olive" title="b2">y</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>] )
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="subset_1.html#S3" title="SUBSET_1:sch.3">SUBSET_1:sch 3</a>();<br/></span>
<span class="kw">hence </span><a NAME="E2:56_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>for <font color="Olive" title="b2">y</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> iff  ex <font color="Olive" title="b3">z</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Olive" title="b3">z</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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