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<div class="add">

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">object</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font> holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a> <font color="Maroon" title="c1">Al</font></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font> implies <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a> <font color="Maroon" title="c1">Al</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:30_1_1_1"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a> <font color="Maroon" title="c1">Al</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">Al</font>, <font color="Maroon" title="c4">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> of <font color="Maroon" title="c1">Al</font><span class="kw"> such that </span><br/><a NAME="E3:30_1_1_1"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<font color="Maroon" title="c2">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="c3">p</font>,<font color="Maroon" title="c4">Sub</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="substut1.html#T8" onmouseover="rs('substut1/T8')" onmouseout="rh()">SUBSTUT1:8</a></span>;<br/><div><span class="lab"><font color="Green" title="E19">A2</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a> <font color="Maroon" title="c1">Al</font></span>)</span> )</span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c5">b</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/><span class="kw">assume </span><a NAME="E1:30_1_1_1_1"/>
 not <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a> <font color="Maroon" title="c1">Al</font></span>)</span></span><br/><a NAME="E2:30_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">Sub</font> <a href="substut1.html#R1" title="SUBSTUT1:pred.1">PQSub</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="substut1.html#D14" onmouseover="rs('substut1/D14')" onmouseout="rh()">SUBSTUT1:def 14</a></span>;<br/><a NAME="E3:30_1_1_1_1"/><span class="kw">then </span>
<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="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">Sub</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a> <font color="Maroon" title="c1">Al</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="substut1.html#D15" onmouseover="rs('substut1/D15')" onmouseout="rh()">SUBSTUT1:def 15</a></span>;<br/><span class="kw">hence </span><a NAME="E4:30_1_1_1_1"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a> <font color="Maroon" title="c1">Al</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a> <font color="Maroon" title="c1">Al</font></span>)</span> )</span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c5">b</font> =  <a href="substut1.html#K14" title="SUBSTUT1:func.14">ExpandSub</a> (<span class="p1">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">bound_in</a> <font color="Maroon" title="c3">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c3">p</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#K21" title="QC_LANG1:func.21">bound_in</a> <font color="Maroon" title="c3">p</font></span>)</span>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">Sub</font>)</span>)</span>);<br/><span class="kw">assume </span><a NAME="E1:30_1_1_1_2"/>
<font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a> <font color="Maroon" title="c1">Al</font></span>)</span></span><br/><a NAME="E2:30_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">Sub</font> <a href="substut1.html#R1" title="SUBSTUT1:pred.1">PQSub</a>  <a href="substut1.html#K14" title="SUBSTUT1:func.14">ExpandSub</a> (<span class="p1">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">bound_in</a> <font color="Maroon" title="c3">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c3">p</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#K21" title="QC_LANG1:func.21">bound_in</a> <font color="Maroon" title="c3">p</font></span>)</span>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">Sub</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="substut1.html#D14" onmouseover="rs('substut1/D14')" onmouseout="rh()">SUBSTUT1:def 14</a></span>;<br/><a NAME="E3:30_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</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="c3">p</font>,<font color="Maroon" title="c4">Sub</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><a href="substut1.html#K14" title="SUBSTUT1:func.14">ExpandSub</a> (<span class="p3">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">bound_in</a> <font color="Maroon" title="c3">p</font></span>)</span>,<span class="p3">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c3">p</font></span>)</span>,<span class="p3">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> (<span class="p4">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">bound_in</a> <font color="Maroon" title="c3">p</font></span>)</span>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">Sub</font>)</span>)</span>)</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a> <font color="Maroon" title="c1">Al</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="substut1.html#D15" onmouseover="rs('substut1/D15')" onmouseout="rh()">SUBSTUT1:def 15</a></span>;<br/><span class="kw">hence </span><a NAME="E4:30_1_1_1_2"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a> <font color="Maroon" title="c1">Al</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E6:30_1_1_1"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a> <font color="Maroon" title="c1">Al</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_1_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<a NAME="E2:30_1_1"/><span class="kw">then </span>
 <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a> <font color="Maroon" title="c1">Al</font></span>)</span>
 
;<br/>
<a NAME="E3:30_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A3</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a> <font color="Maroon" title="c1">Al</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</a></span>;<br/>
<a NAME="E4:30_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a> <font color="Maroon" title="c1">Al</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a> <font color="Maroon" title="c1">Al</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a> <font color="Maroon" title="c1">Al</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font></span>)</span></span>)</span> or <font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a> <font color="Maroon" title="c1">Al</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_1_1_2"/>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a> <font color="Maroon" title="c1">Al</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a> <font color="Maroon" title="c1">Al</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:30_1_1_2"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a> <font color="Maroon" title="c1">Al</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a> <font color="Maroon" title="c1">Al</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E4:30_1_1_2"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<span class="p1">(<span class="default"><a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a> <font color="Maroon" title="c1">Al</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a> <font color="Maroon" title="c1">Al</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T59" onmouseover="rs('relat_1/T59')" onmouseout="rh()">RELAT_1:59</a></span>;<br/>
<a NAME="E5:30_1_1_2"/>
<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="c2">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> <span class="p1">(<span class="default"><a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a> <font color="Maroon" title="c1">Al</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1_2"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">Al</font>, <font color="Maroon" title="c5">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> of <font color="Maroon" title="c1">Al</font>, <font color="Maroon" title="c6">b1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E7:30_1_1_2"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<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="c2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">Sub</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c6">b1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:30_1_1_2"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">Sub</font> <a href="substut1.html#R1" title="SUBSTUT1:pred.1">PQSub</a> <font color="Maroon" title="c6">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_1_2"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="substut1.html#D15" onmouseover="rs('substut1/D15')" onmouseout="rh()">SUBSTUT1:def 15</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E23">A8</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c2">b</font> is   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> of <font color="Maroon" title="c1">Al</font> )</span></a><div class="add"><a NAME="E1:30_1_1_2_1"/><span class="lab"><font color="Green" title="E24">A9</font></span>: 
( <font color="Maroon" title="c6">b1</font> is   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> of <font color="Maroon" title="c1">Al</font> iff <font color="Maroon" title="c6">b1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<span class="p1">(<span class="default"><a href="qc_lang1.html#K3" title="QC_LANG1:func.3">bound_QC-variables</a> <font color="Maroon" title="c1">Al</font></span>)</span>,<span class="p1">(<span class="default"><a href="qc_lang1.html#K3" title="QC_LANG1:func.3">bound_QC-variables</a> <font color="Maroon" title="c1">Al</font></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="substut1.html#D1" onmouseover="rs('substut1/D1')" onmouseout="rh()">SUBSTUT1:def 1</a></span>;<br/><span class="kw">assume </span><a NAME="E2:30_1_1_2_1"/>
 not <font color="Maroon" title="c4">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">b</font> is   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> of <font color="Maroon" title="c1">Al</font></span><br/><a NAME="E3:30_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">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="#E8:30_1_1_2"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="substut1.html#D14" onmouseover="rs('substut1/D14')" onmouseout="rh()">SUBSTUT1:def 14</a></span>;<br/><a NAME="E4:30_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">b1</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K3" title="QC_LANG1:func.3">bound_QC-variables</a> <font color="Maroon" title="c1">Al</font></span>)</span>,<span class="p1">(<span class="default"><a href="qc_lang1.html#K3" title="QC_LANG1:func.3">bound_QC-variables</a> <font color="Maroon" title="c1">Al</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T12" onmouseover="rs('relset_1/T12')" onmouseout="rh()">RELSET_1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E5:30_1_1_2_1"/>
<font color="Maroon" title="c2">b</font> is   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> of <font color="Maroon" title="c1">Al</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1_2_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E7:30_1_1_2"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="partfun1.html#T45" onmouseover="rs('partfun1/T45')" onmouseout="rh()">PARTFUN1:45</a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_1_1_2_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c2">b</font> is   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> of <font color="Maroon" title="c1">Al</font> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:30_1_1_2_2"/>
<font color="Maroon" title="c4">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">b</font> is   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> of <font color="Maroon" title="c1">Al</font></span><br/><a NAME="E2:30_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">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#K21" title="QC_LANG1:func.21">bound_in</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c4">p</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#K21" title="QC_LANG1:func.21">bound_in</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">Sub</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_1_1_2"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="substut1.html#D14" onmouseover="rs('substut1/D14')" onmouseout="rh()">SUBSTUT1:def 14</a></span>;<br/><span class="kw">hence </span><a NAME="E3:30_1_1_2_2"/>
<font color="Maroon" title="c2">b</font> is   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> of <font color="Maroon" title="c1">Al</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1_1_2"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E11:30_1_1_2"/>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a> <font color="Maroon" title="c1">Al</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_1_2"><span class="lab"><font color="Green" title="E23">A8</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E5:30_1_1"/>
<span class="p1">(<span class="default"><a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a> <font color="Maroon" title="c1">Al</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font></span>)</span> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font></span>)</span>,<span class="p1">(<span class="default"><a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a> <font color="Maroon" title="c1">Al</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
