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

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">p</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a> ex <font color="Olive" title="b2">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">Sub</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">Sub</font> <a href="substut1.html#R1" title="SUBSTUT1:pred.1">PQSub</a> $2 );<br/>
<a NAME="E1:23_1_1"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">b2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">b2</font>] holds <br/><font color="Olive" title="b2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b1</font>, <font color="Maroon" title="c3">b2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">b2</font>] implies <font color="Maroon" title="c2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b2</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:23_1_1_1"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
 ex <font color="Olive" title="b1">p</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a> ex <font color="Olive" title="b2">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">Sub</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">Sub</font> <a href="substut1.html#R1" title="SUBSTUT1:pred.1">PQSub</a> <font color="Maroon" title="c2">b1</font> )
 <span class="kw">and </span><br/><a NAME="E2:23_1_1_1"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
 ex <font color="Olive" title="b1">p</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a> ex <font color="Olive" title="b2">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">Sub</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">Sub</font> <a href="substut1.html#R1" title="SUBSTUT1:pred.1">PQSub</a> <font color="Maroon" title="c3">b2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">p1</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a>, <font color="Maroon" title="c5">Sub1</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a><span class="kw"> such that </span><br/><a NAME="E4:23_1_1_1"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">Sub1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:23_1_1_1"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">Sub1</font> <a href="substut1.html#R1" title="SUBSTUT1:pred.1">PQSub</a> <font color="Maroon" title="c2">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1_1"><span class="lab"><font color="Green" title="E7">A2</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">p2</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a>, <font color="Maroon" title="c7">Sub2</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a><span class="kw"> such that </span><br/><a NAME="E7:23_1_1_1"/><span class="lab"><font color="Green" title="E11">A6</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c7">Sub2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:23_1_1_1"/><span class="lab"><font color="Green" title="E12">A7</font></span>: 
<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c7">Sub2</font> <a href="substut1.html#R1" title="SUBSTUT1:pred.1">PQSub</a> <font color="Maroon" title="c3">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1_1_1"><span class="lab"><font color="Green" title="E8">A3</font></span></a></span>;<br/>
<a NAME="E9:23_1_1_1"/><span class="lab"><font color="Green" title="E13">A8</font></span>: 
<font color="Maroon" title="c4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_1_1"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E7:23_1_1_1"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E10:23_1_1_1"/><span class="lab"><font color="Green" title="E14">A9</font></span>: 
<font color="Maroon" title="c5">Sub1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Sub2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_1_1"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E7:23_1_1_1"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:23_1_1_1_1"/>
( <font color="Maroon" title="c4">p1</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  or  not <font color="Maroon" title="c4">p1</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:23_1_1_1_1_1"/><span class="lab"><font color="Green" title="E15">A10</font></span>: 
<font color="Maroon" title="c4">p1</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b2</font></span><br/><div class="add"><a NAME="E2:23_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K14" title="SUBSTUT1:func.14">ExpandSub</a> (<span class="p1">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c4">p1</font></span>)</span>,<span class="p1">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c4">p1</font></span>)</span>,<span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> (<span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c4">p1</font></span>)</span>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">Sub1</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23_1_1_1"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="substut1.html#D14" target="_self" title="SUBSTUT1:def.14">Def14</a></span>;<br/><span class="kw">hence </span><a NAME="E3:23_1_1_1_1_1"/>
<font color="Maroon" title="c2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23_1_1_1"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="txt" href="#E9:23_1_1_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="txt" href="#E10:23_1_1_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="txt" href="#E1:23_1_1_1_1_1"><span class="lab"><font color="Green" title="E15">A10</font></span></a>, <a class="ref" href="substut1.html#D14" target="_self" title="SUBSTUT1:def.14">Def14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:23_1_1_1_1_2"/><span class="lab"><font color="Green" title="E15">A11</font></span>: 
 not <font color="Maroon" title="c4">p1</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b2</font></span><br/><div class="add"><a NAME="E2:23_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23_1_1_1"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="substut1.html#D14" target="_self" title="SUBSTUT1:def.14">Def14</a></span>;<br/><span class="kw">hence </span><a NAME="E3:23_1_1_1_1_2"/>
<font color="Maroon" title="c2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23_1_1_1"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="txt" href="#E9:23_1_1_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="txt" href="#E1:23_1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A11</font></span></a>, <a class="ref" href="substut1.html#D14" target="_self" title="SUBSTUT1:def.14">Def14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c1">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:23_1_1"/><span class="lab"><font color="Green" title="E7">A12</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> iff ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a>,<a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S1" title="FUNCT_1:sch.1">FUNCT_1:sch 1</a>(<a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E6">A1</font></span></a>);<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c1">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> iff  ex <font color="Olive" title="b2">p</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a> ex <font color="Olive" title="b3">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> ex <font color="Olive" title="b4">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">Sub</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">Sub</font> <a href="substut1.html#R1" title="SUBSTUT1:pred.1">PQSub</a> <font color="Olive" title="b4">b</font> ) )</span><br/>

<a NAME="E4:23_1_1"/>
 for <font color="Olive" title="b1">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> iff  ex <font color="Olive" title="b2">p</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a> ex <font color="Olive" title="b3">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> ex <font color="Olive" title="b4">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">Sub</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">Sub</font> <a href="substut1.html#R1" title="SUBSTUT1:pred.1">PQSub</a> <font color="Olive" title="b4">b</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> iff  ex <font color="Olive" title="b1">p</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a> ex <font color="Olive" title="b2">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> ex <font color="Olive" title="b3">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">Sub</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">Sub</font> <a href="substut1.html#R1" title="SUBSTUT1:pred.1">PQSub</a> <font color="Olive" title="b3">b</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:23_1_1_2"/>
( <font color="Maroon" title="c2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> implies  ex <font color="Olive" title="b1">p</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a> ex <font color="Olive" title="b2">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> ex <font color="Olive" title="b3">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">Sub</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">Sub</font> <a href="substut1.html#R1" title="SUBSTUT1:pred.1">PQSub</a> <font color="Olive" title="b3">b</font> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">p</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a> ex <font color="Olive" title="b2">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> ex <font color="Olive" title="b3">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">Sub</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">Sub</font> <a href="substut1.html#R1" title="SUBSTUT1:pred.1">PQSub</a> <font color="Olive" title="b3">b</font> ) implies <font color="Maroon" title="c2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:23_1_1_2_1"/><span class="lab"><font color="Green" title="E8">A13</font></span>: 
<font color="Maroon" title="c2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a> ex <font color="Olive" title="b2">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> ex <font color="Olive" title="b3">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">Sub</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">Sub</font> <a href="substut1.html#R1" title="SUBSTUT1:pred.1">PQSub</a> <font color="Olive" title="b3">b</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:23_1_1_2_1"/><span class="lab"><font color="Green" title="E9">A14</font></span>: 
<font color="Maroon" title="c2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/>
<a NAME="E4:23_1_1_2_1"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a> ex <font color="Olive" title="b2">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> st <br/>( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">Sub</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">Sub</font> <a href="substut1.html#R1" title="SUBSTUT1:pred.1">PQSub</a> <font color="Maroon" title="c4">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E7">A12</font></span></a>, <a class="txt" href="#E1:23_1_1_2_1"><span class="lab"><font color="Green" title="E8">A13</font></span></a>, <a class="txt" href="#E3:23_1_1_2_1"><span class="lab"><font color="Green" title="E9">A14</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:23_1_1_2_1"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a> ex <font color="Olive" title="b2">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> ex <font color="Olive" title="b3">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">Sub</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">Sub</font> <a href="substut1.html#R1" title="SUBSTUT1:pred.1">PQSub</a> <font color="Olive" title="b3">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_1_2_1"><span class="lab"><font color="Green" title="E9">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:23_1_1_2"/>
(  ex <font color="Olive" title="b1">p</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a> ex <font color="Olive" title="b2">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> ex <font color="Olive" title="b3">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">Sub</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">Sub</font> <a href="substut1.html#R1" title="SUBSTUT1:pred.1">PQSub</a> <font color="Olive" title="b3">b</font> ) implies <font color="Maroon" title="c2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E7">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:23_1_1"/>
 for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> iff  ex <font color="Olive" title="b2">p</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a> ex <font color="Olive" title="b3">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> ex <font color="Olive" title="b4">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">Sub</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">Sub</font> <a href="substut1.html#R1" title="SUBSTUT1:pred.1">PQSub</a> <font color="Olive" title="b4">b</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
