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<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>



<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> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>( <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  =  <a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">a</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>( <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = 0;<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:39"/><i><font color="Green">E27</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds <br/> ( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>) ) ) ) )
 <b>from </b><i><a class="ref" href="partfun1.html#S6">PARTFUN1:sch 6</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i>, </i>;<br/><div><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/><a NAME="E2:39_1_1"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">a</font>,<font color="Maroon">x</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:39_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/><a NAME="E2:39_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:39_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:39_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="integra6.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:39"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a>  <b>by </b><i>, <a class="ref" href="partfun1.html#T25">PARTFUN1:25</a></i>;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E6:39"/>
( <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; ( for <font color="Olive">x</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">a</font>,<font color="Olive">x</font> ) )
 <b>by </b><i/>;<br/>


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