<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <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">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a> </span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:135_1_2"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="seqm_3.html#V6">natural-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> )
 <b>and </b><br/><a NAME="E2:135_1_2"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="seqm_3.html#V6">natural-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> iff for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">b<sub>2</sub></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">c<sub>5</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:135_1_2_1"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#T169">FUNCT_2:169</a></i>;<br/><a NAME="E2:135_1_2_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is  <a href="seqm_3.html#V6">natural-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">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">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:135_1_2_1_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="polynom1.html#E1:135_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="polynom1.html#E2:135_1_2_1"><i><font color="Green">E75</font></i></a></i>;<br/><b>hence </b><a NAME="E3:135_1_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="polynom1.html#E2:135_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="polynom1.html#E2:135_1_2_1"><i><font color="Green">E75</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E4:135_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E5:135_1_2_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="polynom1.html#E2:135_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="polynom1.html#E2:135_1_2_1"><i><font color="Green">E75</font></i></a></i>;<br/><b>hence </b><a NAME="E6:135_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="polynom1.html#E1:135_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="polynom1.html#E2:135_1_2_1"><i><font color="Green">E75</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:135_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="subset_1.html#T8">SUBSET_1:8</a></i>;<br/>


</div>
