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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="real_1.html#NM1">Real</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = 1 <a href="real_1.html#K4">*</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E2:10_1_1"/><i><font color="Green">E44</font></i>: 
( ( for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>] ) ) &amp; ( for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">r</font>) ) )
 <b>from </b><i><a class="ref" href="seq_1.html#S3">SEQ_1:sch 3</a>();<br/></i>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<a NAME="E3:10_1_1"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  holds <br/><font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:10_1_1"/><b>then </b>
 <a href="numbers.html#K1">REAL</a>  <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:10_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E6:10_1_1"/>
<font color="Maroon">f</font> is <a href="partfun1.html#V1">total</a>
 <b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="fdiff_1.html#D4" target="_self">FDIFF_1:def 4</a><br/>

<a NAME="E7:10_1_1"/>
for <font color="Olive">p</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K4">*</a> <font color="Olive">p</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:10_1_1"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>for <font color="Olive">p</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="real_1.html#K4">*</a> <font color="Olive">p</font>
 ;<br/>


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