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

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ] <span class="kw">means </span><font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1;<br/>
<a NAME="E1:75_1"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">P</font> being  <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Olive" title="b1">k</font><br/> for <font color="Olive" title="b3">l</font> being  <a href="qc_lang1.html#NM7" title="QC_LANG1:NM.7">QC-variable_list</a> of <font color="Olive" title="b1">k</font><br/> for <font color="Olive" title="b4">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a>  holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Olive" title="b2">P</font>,<font color="Olive" title="b3">l</font>,<font color="Olive" title="b4">e</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font> being  <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c1">k</font><br/> for <font color="Olive" title="b2">l</font> being  <a href="qc_lang1.html#NM7" title="QC_LANG1:NM.7">QC-variable_list</a> of <font color="Maroon" title="c1">k</font><br/> for <font color="Olive" title="b3">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a>  holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">l</font>,<font color="Olive" title="b3">e</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c2">P</font> be   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c1">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">l</font> being  <a href="qc_lang1.html#NM7" title="QC_LANG1:NM.7">QC-variable_list</a> of <font color="Maroon" title="c1">k</font><br/> for <font color="Olive" title="b2">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a>  holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Olive" title="b1">l</font>,<font color="Olive" title="b2">e</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c3">l</font> be   <a href="qc_lang1.html#NM7" title="QC_LANG1:NM.7">QC-variable_list</a> of <font color="Maroon" title="c1">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a>  holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">l</font>,<font color="Olive" title="b1">e</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c4">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">e</font>]</span><br/>







<span class="kw">thus </span><font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">e</font></span>)</span> = 
<font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">e</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E43">A1</font></span></a></span>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">e</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E44">A2</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:75_1"/><span class="lab"><font color="Green" title="E46">A4</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/> for <font color="Olive" title="b2">S</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> <br/> for <font color="Olive" title="b3">SQ</font> being   <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Olive" title="b2">S</font>,<font color="Olive" title="b1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>  st <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Olive" title="b2">S</font>,<font color="Olive" title="b1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">S</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Olive" title="b2">S</font>,<font color="Olive" title="b1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>,<font color="Olive" title="b3">SQ</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">x</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> <br/> for <font color="Olive" title="b2">SQ</font> being   <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>  st <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">S</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>,<font color="Olive" title="b2">SQ</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">SQ</font> being   <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>  st <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">S</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>,<font color="Olive" title="b1">SQ</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c3">SQ</font> be    <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">S</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>,<font color="Maroon" title="c3">SQ</font>] )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:75_1_2"/><span class="lab"><font color="Green" title="E47">A5</font></span>: 
<span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a> 
 <span class="kw">and </span><br/><a NAME="E2:75_1_2"/><span class="lab"><font color="Green" title="E48">A6</font></span>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>,<font color="Maroon" title="c3">SQ</font>]</span><br/>

<a NAME="E3:75_1_2"/><span class="lab"><font color="Green" title="E49">A7</font></span>: 
 <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>,<font color="Maroon" title="c3">SQ</font> is  <a href="substut1.html#V7" title="SUBSTUT1:attr.7">Sub_universal</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1_2"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="ref" href="substut1.html#T14" target="_self" title="SUBSTUT1:th.14">Th14</a></span>;<br/>
<a NAME="E4:75_1_2"/>
 <a href="substut1.html#K31" title="SUBSTUT1:func.31">Sub_the_scope_of</a> <span class="p1">(<span class="default"><a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> <span class="p2"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>,<font color="Maroon" title="c3">SQ</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span> <a href="substut1.html#K22" title="SUBSTUT1:func.22">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1_2"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="ref" href="substut1.html#T21" target="_self" title="SUBSTUT1:th.21">Th21</a></span>;<br/>
<a NAME="E5:75_1_2"/><span class="kw">then </span>
 <a href="substut1.html#K31" title="SUBSTUT1:func.31">Sub_the_scope_of</a> <span class="p1">(<span class="default"><a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> <span class="p2"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>,<font color="Maroon" title="c3">SQ</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> <span class="p2"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>,<font color="Maroon" title="c3">SQ</font></span>)</span> = 
<font color="Maroon">F<sub>8</sub></font>(<span class="p1">(<span class="default"><a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> <span class="p2"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>,<font color="Maroon" title="c3">SQ</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K31" title="SUBSTUT1:func.31">Sub_the_scope_of</a> <span class="p3">(<span class="default"><a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> <span class="p4"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>,<font color="Maroon" title="c3">SQ</font></span>)</span></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E2:75_1_2"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="txt" href="#E3:75_1_2"><span class="lab"><font color="Green" title="E49">A7</font></span></a></span>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> <span class="p2"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>,<font color="Maroon" title="c3">SQ</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="#E3:75_1_2"><span class="lab"><font color="Green" title="E49">A7</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:75_1"/><span class="lab"><font color="Green" title="E47">A8</font></span>: 
for <font color="Olive" title="b1">S</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a>   st <font color="Olive" title="b1">S</font> is  <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">S</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">S</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">S</font> is  <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a>  implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">S</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:75_1_3"/><span class="lab"><font color="Green" title="E48">A9</font></span>: 
<font color="Maroon" title="c1">S</font> is  <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">S</font>]</span><br/>

<a NAME="E2:75_1_3"/><span class="kw">then </span>
<font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E43">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:75_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">S</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="#E1:75_1_3"><span class="lab"><font color="Green" title="E48">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:75_1"/><span class="lab"><font color="Green" title="E48">A10</font></span>: 
for <font color="Olive" title="b1">S1</font>, <font color="Olive" title="b2">S2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a>   st <font color="Olive" title="b1">S1</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">S2</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">S1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">S2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K21" title="SUBSTUT1:func.21">Sub_&amp;</a> <font color="Olive" title="b1">S1</font>,<font color="Olive" title="b2">S2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">S2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">S1</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S2</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">S1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">S2</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K21" title="SUBSTUT1:func.21">Sub_&amp;</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:75_1_4"/><span class="lab"><font color="Green" title="E49">A11</font></span>: 
<font color="Maroon" title="c1">S1</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S2</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> 
 <span class="kw">and </span><br/><a NAME="E2:75_1_4"/><span class="lab"><font color="Green" title="E50">A12</font></span>: 
( <font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">S1</font> &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">S2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K21" title="SUBSTUT1:func.21">Sub_&amp;</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font>]</span><br/>

<a NAME="E3:75_1_4"/><span class="lab"><font color="Green" title="E51">A13</font></span>: 
 <a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <span class="p1">(<span class="default"><a href="substut1.html#K21" title="SUBSTUT1:func.21">Sub_&amp;</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1_4"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="ref" href="substut1.html#T19" target="_self" title="SUBSTUT1:th.19">Th19</a></span>;<br/>
<a NAME="E4:75_1_4"/><span class="lab"><font color="Green" title="E52">A14</font></span>: 
(  <a href="substut1.html#K21" title="SUBSTUT1:func.21">Sub_&amp;</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font> is  <a href="substut1.html#V6" title="SUBSTUT1:attr.6">Sub_conjunctive</a>  &amp;  <a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <span class="p1">(<span class="default"><a href="substut1.html#K21" title="SUBSTUT1:func.21">Sub_&amp;</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1_4"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="ref" href="substut1.html#T13" target="_self" title="SUBSTUT1:th.13">Th13</a>, <a class="ref" href="substut1.html#T18" target="_self" title="SUBSTUT1:th.18">Th18</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K21" title="SUBSTUT1:func.21">Sub_&amp;</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> = 
<font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">S1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">S2</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E2:75_1_4"><span class="lab"><font color="Green" title="E50">A12</font></span></a>, <a class="txt" href="#E3:75_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a></span>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K21" title="SUBSTUT1:func.21">Sub_&amp;</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="#E4:75_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="txt" href="#E3:75_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:75_1"/><span class="lab"><font color="Green" title="E49">A15</font></span>: 
for <font color="Olive" title="b1">S</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">S</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K20" title="SUBSTUT1:func.20">Sub_not</a> <font color="Olive" title="b1">S</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">S</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">S</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K20" title="SUBSTUT1:func.20">Sub_not</a> <font color="Maroon" title="c1">S</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:75_1_5"/><span class="lab"><font color="Green" title="E50">A16</font></span>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K20" title="SUBSTUT1:func.20">Sub_not</a> <font color="Maroon" title="c1">S</font>]</span><br/>

<a NAME="E2:75_1_5"/><span class="lab"><font color="Green" title="E51">A17</font></span>: 
 <a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <span class="p1">(<span class="default"><a href="substut1.html#K20" title="SUBSTUT1:func.20">Sub_not</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="substut1.html#D30" target="_self" title="SUBSTUT1:def.30">Def30</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K20" title="SUBSTUT1:func.20">Sub_not</a> <font color="Maroon" title="c1">S</font></span>)</span> = 
<font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">S</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E1:75_1_5"><span class="lab"><font color="Green" title="E50">A16</font></span></a></span>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K20" title="SUBSTUT1:func.20">Sub_not</a> <font color="Maroon" title="c1">S</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="#E2:75_1_5"><span class="lab"><font color="Green" title="E51">A17</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:75_1"/>
for <font color="Olive" title="b1">S</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">S</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="substut1.html#S1" target="_self" title="SUBSTUT1:sch.1">SUBSTUT1:sch 1</a>(<a class="txt" href="#E1:75_1"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E3:75_1"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="txt" href="#E5:75_1"><span class="lab"><font color="Green" title="E49">A15</font></span></a>, <a class="txt" href="#E4:75_1"><span class="lab"><font color="Green" title="E48">A10</font></span></a>, <a class="txt" href="#E2:75_1"><span class="lab"><font color="Green" title="E46">A4</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E7:75_1"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">F<sub>3</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
