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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means for <font color="Olive">b<sub>1</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>2</sub></font>();<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() = <font color="Maroon">F<sub>5</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>3</sub></font>();<br/>
<a NAME="E1:43_1"/><i><font color="Green">E29</font></i>: 
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>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() ) )
 
<b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>1</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:43_1"/><i><font color="Green">E30</font></i>: 
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>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="funct_2.html#S5">FUNCT_2:sch 5</a>(<a class="txt" href="valuat_1.html#E1:43_1"><i><font color="Green">E29</font></i></a>);<br/></i>
<a NAME="E4:43_1"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>() )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E5:43_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>2</sub></font>()
 ;<br/><a NAME="E2:43_1_1_1"/>
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>2</sub></font>() ) implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="valuat_1.html#E3:43_1"><i><font color="Green">E30</font></i></a></i>;<br/><b>hence </b><a NAME="E3:43_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:43_1_1_1"><i><font color="Green">E31</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:43_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>2</sub></font>() implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>thus </b><a NAME="E3:43_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>3</sub></font>() )
 <b>by </b><i><a class="txt" href="valuat_1.html#E3:43_1"><i><font color="Green">E30</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:43_1"/>
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>3</sub></font>() )
 ;<br/>


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