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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> ex <font color="Olive">g</font> being <a href="seqm_3.html#V6">natural-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">g</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">x</font> ) );<br/>
<b>consider </b><font color="Maroon">IT</font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">X</font>,<a href="numbers.html#K5">NAT</a> </span>)</span><b> such that </b><br/><a NAME="E2:135_1_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">h</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> iff ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">X</font>,<a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">h</font>] ) )
 <b>from </b><i><a class="ref" href="subset_1.html#S1">SUBSET_1:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">IT</font>
;<br/>

<b>let </b><font color="Maroon">g</font> be  <a href="seqm_3.html#V6">natural-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">X</font>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:135_1_1_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font>
 ;<br/><b>then consider </b><font color="Maroon">g1</font> being  <a href="seqm_3.html#V6">natural-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:135_1_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">g1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">g</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">g1</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">x</font> ) )
 <b>by </b><i><a class="ref" href="polynom1.html#T9" target="_self">Th9</a></i>;<br/><b>thus </b><a NAME="E4:135_1_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">g</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">x</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:135_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">g</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">x</font>
 ;<br/>

<a NAME="E5:135_1_1"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a>, <a class="ref" href="seqm_3.html#D8">SEQM_3:def 8</a></i>;<br/>
<a NAME="E6:135_1_1"/><b>then </b>
<font color="Maroon">g</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>, <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E7:135_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">X</font>,<a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>hence </b><a NAME="E8:135_1_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T9" target="_self">Th9</a>, </i>;<br/>


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