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

<div><span class="lab"><font color="Green" title="E6">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c4">Z</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="relat_1.html#V1" title="RELAT_1:attr.1">Relation-like</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><span class="kw">assume </span><a NAME="E2:21_1_1_1"/>
 not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">finSub</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="c3">finSub</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a></span><br/><div><span class="lab"><font color="Green" title="E7">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c5">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a><span class="kw"> such that </span><br/><a NAME="E2:21_1_1_1_1"/><span class="lab"><font color="Green" title="E8">A4</font></span>: 
<font color="Maroon" title="c3">finSub</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> (<font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span>,<font color="Maroon" title="c5">Sub</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c6">X</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> &amp; <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">Sub</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">Sub</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) </span> } </span> ;<br/><a NAME="E3:21_1_1_1_1"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">finSub</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> &amp; <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">Sub</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">Sub</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_1_1_1"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="ref" href="relat_1.html#T58" title="RELAT_1:th.58">RELAT_1:58</a></span>;<br/><span class="kw">given </span><font color="Maroon" title="c7">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><a NAME="E5:21_1_1_1_1"/><span class="lab"><font color="Green" title="E10">A6</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">finSub</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">Z</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E6:21_1_1_1_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">finSub</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_1_1_1_1"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E7:21_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> &amp; <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">Sub</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">Sub</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1_1_1_1"><span class="lab"><font color="Green" title="E9">A5</font></span></a></span>;<br/><a NAME="E8:21_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A7</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  ex <font color="Olive" title="b1">y</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> st <br/>( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> &amp; <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">Sub</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">Sub</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T9" title="RELAT_1:th.9">RELAT_1:9</a></span>;<br/><a NAME="E9:21_1_1_1_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_1_1_1_1"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E10:21_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_1_1_1_1"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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">reconsider </span><font color="Maroon" title="c5">Z</font> = <font color="Maroon" title="c4">Z</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/><a NAME="E5:21_1_1_1"/>
 for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">finSub</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">Z</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">finSub</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1_1_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E7:21_1_1_1"/><span class="lab"><font color="Green" title="E8">A8</font></span>: 
<span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">finSub</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T1" title="PARTFUN1:th.1">PARTFUN1:1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">Z</font> = <font color="Maroon" title="c5">Z</font> as   <a href="relset_1.html#NM1" title="RELSET_1:NM.1">Relation</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ;<br/><a NAME="E9:21_1_1_1"/>
<span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">finSub</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_1_1_1"><span class="lab"><font color="Green" title="E8">A8</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:21_1_1_1"/>
<font color="Maroon" title="c3">finSub</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T45" title="PARTFUN1:th.45">PARTFUN1:45</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="hs2(this)" href="javascript:()" title="21_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c4">Z</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <span class="p4">(<span class="default"><a href="substut1.html#K13" title="SUBSTUT1:func.13">upVar</a> (<font color="Maroon" title="c3">finSub</font>,<font color="Maroon" title="c2">p</font>)</span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="relat_1.html#V1" title="RELAT_1:attr.1">Relation-like</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><span class="kw">assume </span><a NAME="E2:21_1_1_2"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">finSub</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="c3">finSub</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <span class="p4">(<span class="default"><a href="substut1.html#K13" title="SUBSTUT1:func.13">upVar</a> (<font color="Maroon" title="c3">finSub</font>,<font color="Maroon" title="c2">p</font>)</span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a></span><br/><div><span class="lab"><font color="Green" title="E7">A9</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c5">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a><span class="kw"> such that </span><br/><a NAME="E2:21_1_1_2_1"/><span class="lab"><font color="Green" title="E8">A10</font></span>: 
<font color="Maroon" title="c3">finSub</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> (<font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span>,<font color="Maroon" title="c5">Sub</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c6">X</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> &amp; <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">Sub</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">Sub</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) </span> } </span> ;<br/><a NAME="E3:21_1_1_2_1"/><span class="lab"><font color="Green" title="E9">A11</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">finSub</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> &amp; <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">Sub</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">Sub</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_1_2_1"><span class="lab"><font color="Green" title="E8">A10</font></span></a>, <a class="ref" href="relat_1.html#T58" title="RELAT_1:th.58">RELAT_1:58</a></span>;<br/><span class="kw">given </span><font color="Maroon" title="c7">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><a NAME="E5:21_1_1_2_1"/><span class="lab"><font color="Green" title="E10">A12</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">finSub</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">Z</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E6:21_1_1_2_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">finSub</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_1_1_2_1"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E7:21_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> &amp; <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">Sub</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">Sub</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1_1_2_1"><span class="lab"><font color="Green" title="E9">A11</font></span></a></span>;<br/><a NAME="E8:21_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A13</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  ex <font color="Olive" title="b1">y</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> st <br/>( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> &amp; <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">Sub</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">Sub</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T9" title="RELAT_1:th.9">RELAT_1:9</a></span>;<br/><a NAME="E9:21_1_1_2_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_1_1_2_1"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E10:21_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_1_1_2_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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">reconsider </span><font color="Maroon" title="c5">Z</font> = <font color="Maroon" title="c4">Z</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/><a NAME="E5:21_1_1_2"/>
 for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">finSub</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">Z</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">finSub</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1_1_2"><span class="lab"><font color="Green" title="E7">A9</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E7:21_1_1_2"/><span class="lab"><font color="Green" title="E8">A14</font></span>: 
<span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">finSub</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T1" title="PARTFUN1:th.1">PARTFUN1:1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">Z</font> = <font color="Maroon" title="c5">Z</font> as   <a href="relset_1.html#NM1" title="RELSET_1:NM.1">Relation</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ;<br/><a NAME="E9:21_1_1_2"/>
<span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">finSub</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_1_1_2"><span class="lab"><font color="Green" title="E8">A14</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:21_1_1_2"/>
<font color="Maroon" title="c3">finSub</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <span class="p4">(<span class="default"><a href="substut1.html#K13" title="SUBSTUT1:func.13">upVar</a> (<font color="Maroon" title="c3">finSub</font>,<font color="Maroon" title="c2">p</font>)</span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T45" title="PARTFUN1:th.45">PARTFUN1:45</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="E3:21_1_1"/>
( ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">finSub</font> implies <font color="Maroon" title="c3">finSub</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <span class="p4">(<span class="default"><a href="substut1.html#K13" title="SUBSTUT1:func.13">upVar</a> (<font color="Maroon" title="c3">finSub</font>,<font color="Maroon" title="c2">p</font>)</span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> ) &amp; (  not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">finSub</font> implies <font color="Maroon" title="c3">finSub</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1"><span class="lab"><font color="Green" title="E6">A2</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>
