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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="qc_lang1.html#NM8">QC-formula</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="substut1.html#NM1">CQC_Substitution</a> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="substut1.html#R1" target="_self">PQSub</a> <font color="Maroon">a<sub>2</sub></font> );<br/>
<a NAME="E1:23_1_1"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:23_1_1_1"/><i><font color="Green">E7</font></i>: 
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="qc_lang1.html#NM8">QC-formula</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="substut1.html#NM1">CQC_Substitution</a> st <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="substut1.html#R1" target="_self">PQSub</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="qc_lang1.html#NM8">QC-formula</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="substut1.html#NM1">CQC_Substitution</a> st <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="substut1.html#R1" target="_self">PQSub</a> <font color="Maroon">c<sub>3</sub></font> ) )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="qc_lang1.html#NM8">QC-formula</a>, <font color="Maroon">c<sub>5</sub></font> being   <a href="substut1.html#NM1">CQC_Substitution</a><b> such that </b><br/><a NAME="E3:23_1_1_1"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="substut1.html#R1" target="_self">PQSub</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="substut1.html#E1:23_1_1_1"><i><font color="Green">E7</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="qc_lang1.html#NM8">QC-formula</a>, <font color="Maroon">c<sub>7</sub></font> being   <a href="substut1.html#NM1">CQC_Substitution</a><b> such that </b><br/><a NAME="E5:23_1_1_1"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="substut1.html#R1" target="_self">PQSub</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="substut1.html#E1:23_1_1_1"><i><font color="Green">E7</font></i></a></i>;<br/>
<a NAME="E6:23_1_1_1"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="substut1.html#E3:23_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="substut1.html#E5:23_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V5">universal</a> or  not <font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V5">universal</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V5">universal</a>
 ;<br/><div class="add"><a NAME="E2:23_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K14" target="_self">ExpandSub</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="substut1.html#K7" target="_self">RestrictSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K14" target="_self">ExpandSub</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="substut1.html#K7" target="_self">RestrictSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="substut1.html#E3:23_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="substut1.html#E5:23_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="substut1.html#E6:23_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="substut1.html#D14" target="_self">Def14</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="substut1.html#E6:23_1_1_1"><i><font color="Green">E10</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_1_1_2"/>
not <font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V5">universal</a>
 ;<br/><div class="add"><a NAME="E2:23_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="substut1.html#E3:23_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="substut1.html#E5:23_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="substut1.html#E6:23_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="substut1.html#D14" target="_self">Def14</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>1</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:23_1_1"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="qc_lang1.html#K8">QC-WFF</a> ,<a href="substut1.html#K1" target="_self">vSUB</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S1">FUNCT_1:sch 1</a>(<a class="txt" href="substut1.html#E1:23_1_1"><i><font color="Green">E6</font></i></a>);<br/></i>
<a NAME="E4:23_1_1"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> iff ex <font color="Olive">b<sub>2</sub></font> being  <a href="qc_lang1.html#NM8">QC-formula</a>ex <font color="Olive">b<sub>3</sub></font> being  <a href="substut1.html#NM1">CQC_Substitution</a>ex <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="substut1.html#R1" target="_self">PQSub</a> <font color="Olive">b<sub>4</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:23_1_1_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="qc_lang1.html#NM8">QC-formula</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="substut1.html#NM1">CQC_Substitution</a>ex <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="substut1.html#R1" target="_self">PQSub</a> <font color="Olive">b<sub>3</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_1_2_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:23_1_1_2_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="qc_lang1.html#NM8">QC-formula</a>, <font color="Maroon">c<sub>6</sub></font> being   <a href="substut1.html#NM1">CQC_Substitution</a><b> such that </b><br/><a NAME="E5:23_1_1_2_1"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="substut1.html#R1" target="_self">PQSub</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="substut1.html#E3:23_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="substut1.html#E1:23_1_1_2_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="substut1.html#E3:23_1_1_2_1"><i><font color="Green">E10</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:23_1_1_2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="qc_lang1.html#NM8">QC-formula</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="substut1.html#NM1">CQC_Substitution</a>ex <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="substut1.html#R1" target="_self">PQSub</a> <font color="Olive">b<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="substut1.html#E3:23_1_1_2_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="substut1.html#E5:23_1_1_2_1"><i><font color="Green">E11</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:23_1_1_2"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="qc_lang1.html#NM8">QC-formula</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="substut1.html#NM1">CQC_Substitution</a>ex <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="substut1.html#R1" target="_self">PQSub</a> <font color="Olive">b<sub>3</sub></font> ) implies <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="substut1.html#E3:23_1_1"><i><font color="Green">E7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">c<sub>1</sub></font>
;<br/>

<b>thus </b><a NAME="E5:23_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> iff ex <font color="Olive">b<sub>2</sub></font> being  <a href="qc_lang1.html#NM8">QC-formula</a>ex <font color="Olive">b<sub>3</sub></font> being  <a href="substut1.html#NM1">CQC_Substitution</a>ex <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="substut1.html#R1" target="_self">PQSub</a> <font color="Olive">b<sub>4</sub></font> ) )
 <b>by </b><i><a class="txt" href="substut1.html#E4:23_1_1"><i><font color="Green">E8</font></i></a></i>;<br/>


</div>
