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<b>let </b><font color="Maroon" title="c3">it1</font>, <font color="Maroon" title="c4">it2</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:135_1_2"/><i><font color="Green" title="E73">A4</font></i>: 
for <font color="Olive" title="b1">g</font> being <a href="seqm_3.html#V6" title="SEQM_3:attr.6">natural-yielding</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">it1</font> iff for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Olive" title="b1">g</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b2">x</font> )
 <b>and </b><br/><a NAME="E2:135_1_2"/><i><font color="Green" title="E74">A5</font></i>: 
for <font color="Olive" title="b1">g</font> being <a href="seqm_3.html#V6" title="SEQM_3:attr.6">natural-yielding</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">it2</font> iff for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Olive" title="b1">g</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b2">x</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="135_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">h</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:135_1_2_1"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#T169" title="FUNCT_2:th.169">FUNCT_2:169</a></i>;<br/><a NAME="E2:135_1_2_1"/><b>then </b><i><font color="Green" title="E75">A6</font></i>: 
<font color="Maroon" title="c5">h</font> is  <a href="seqm_3.html#V6" title="SEQM_3:attr.6">natural-yielding</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:135_1_2_1_1"/>
<font color="Maroon" title="c5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">it1</font>
 ;<br/><a NAME="E2:135_1_2_1_1"/><b>then </b>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c5">h</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E1:135_1_2"><i><font color="Green" title="E73">A4</font></i></a>, <a class="txt" href="polynom1.html#E2:135_1_2_1"><i><font color="Green" title="E75">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E3:135_1_2_1_1"/>
<font color="Maroon" title="c5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">it2</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E2:135_1_2"><i><font color="Green" title="E74">A5</font></i></a>, <a class="txt" href="polynom1.html#E2:135_1_2_1"><i><font color="Green" title="E75">A6</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E4:135_1_2_1"/>
<font color="Maroon" title="c5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">it2</font>
 ;<br/><a NAME="E5:135_1_2_1"/><b>then </b>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c5">h</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E2:135_1_2"><i><font color="Green" title="E74">A5</font></i></a>, <a class="txt" href="polynom1.html#E2:135_1_2_1"><i><font color="Green" title="E75">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E6:135_1_2_1"/>
<font color="Maroon" title="c5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">it1</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E1:135_1_2"><i><font color="Green" title="E73">A4</font></i></a>, <a class="txt" href="polynom1.html#E2:135_1_2_1"><i><font color="Green" title="E75">A6</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:135_1_2"/>
<font color="Maroon" title="c3">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">it2</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T8" title="SUBSET_1:th.8">SUBSET_1:8</a></i>;<br/>


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