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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(),  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ,<font color="Maroon">F<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> ] <b>means </b>( ( $3 is <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a> implies $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( $3 is <a href="substut1.html#V4" title="SUBSTUT1:attr.4">Sub_atomic</a> implies $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>($3) ) &amp; ( $3 is <a href="substut1.html#V5" title="SUBSTUT1:attr.5">Sub_negative</a> implies $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default">$2 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> $3</span>)</span></span>)</span>) ) &amp; ( $3 is <a href="substut1.html#V6" title="SUBSTUT1:attr.6">Sub_conjunctive</a> implies $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default">$2 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> $3</span>)</span></span>)</span>,<span class="p1">(<span class="default">$2 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> $3</span>)</span></span>)</span>) ) &amp; ( $3 is <a href="substut1.html#V7" title="SUBSTUT1:attr.7">Sub_universal</a> implies $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>($3,<span class="p1">(<span class="default">$2 <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> $3</span>)</span></span>)</span>) ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>(),   <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> ] <b>means </b>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  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $2 holds <br/>( ( <font color="Olive" title="b1">S</font> is <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a> implies $1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( <font color="Olive" title="b1">S</font> is <a href="substut1.html#V4" title="SUBSTUT1:attr.4">Sub_atomic</a> implies $1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">S</font>) ) &amp; ( <font color="Olive" title="b1">S</font> is <a href="substut1.html#V5" title="SUBSTUT1:attr.5">Sub_negative</a> implies $1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default">$1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Olive" title="b1">S</font></span>)</span></span>)</span>) ) &amp; ( <font color="Olive" title="b1">S</font> is <a href="substut1.html#V6" title="SUBSTUT1:attr.6">Sub_conjunctive</a> implies $1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default">$1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Olive" title="b1">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default">$1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Olive" title="b1">S</font></span>)</span></span>)</span>) ) &amp; ( <font color="Olive" title="b1">S</font> is <a href="substut1.html#V7" title="SUBSTUT1:attr.7">Sub_universal</a> implies $1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive" title="b1">S</font>,<span class="p1">(<span class="default">$1 <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> <font color="Olive" title="b1">S</font></span>)</span></span>)</span>) ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <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> ] <b>means </b> ex <font color="Olive" title="b1">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">F</font>,$1];<br/>
<a NAME="E1:74_1"/><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c1">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>();<br/>
<b>take </b>
<font color="Maroon" title="c1">F</font>
;<br/>

<b>let </b><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> ;<br/>

<b>assume </b><a NAME="E2:74_1_1"/><i><font color="Green" title="E46">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<b>thus </b><a NAME="E3:74_1_1"/>
( ( <font color="Maroon" title="c2">S</font> is <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a> implies <font color="Maroon" title="c1">F</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>2</sub></font>() ) &amp; ( <font color="Maroon" title="c2">S</font> is <a href="substut1.html#V4" title="SUBSTUT1:attr.4">Sub_atomic</a> implies <font color="Maroon" title="c1">F</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>(<font color="Maroon" title="c2">S</font>) ) &amp; ( <font color="Maroon" title="c2">S</font> is <a href="substut1.html#V5" title="SUBSTUT1:attr.5">Sub_negative</a> implies <font color="Maroon" title="c1">F</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>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c2">S</font> is <a href="substut1.html#V6" title="SUBSTUT1:attr.6">Sub_conjunctive</a> implies <font color="Maroon" title="c1">F</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>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c2">S</font> is <a href="substut1.html#V7" title="SUBSTUT1:attr.7">Sub_universal</a> implies <font color="Maroon" title="c1">F</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>6</sub></font>(<font color="Maroon" title="c2">S</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</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> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>) ) )
 <b>by </b><i><a class="txt" href="substut1.html#E2:74_1_1"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="qc_lang1.html#T34" title="QC_LANG1:th.34">QC_LANG1:34</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:74_1"/><i><font color="Green" title="E46">A3</font></i>: 
for <font color="Olive" title="b1">n</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>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">n</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> ;<br/>

<b>given </b><font color="Maroon" title="c2">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><a NAME="E2:74_1_2"/><i><font color="Green" title="E47">A4</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">n</font>]
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>4</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> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()] <b>means </b>( (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default">$1 <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 ) &amp; (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default">$1 <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <font color="Maroon">S<sub>1</sub></font>[$2,<font color="Maroon" title="c2">F</font>,$1] ) );<br/>
<a NAME="E3:74_1_2"/><i><font color="Green" title="E48">A5</font></i>: 
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>   ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">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> ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_1_2_1_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c3">S</font> is <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a> ) or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c3">S</font> is <a href="substut1.html#V4" title="SUBSTUT1:attr.4">Sub_atomic</a> ) or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c3">S</font> is <a href="substut1.html#V5" title="SUBSTUT1:attr.5">Sub_negative</a> ) or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c3">S</font> is <a href="substut1.html#V6" title="SUBSTUT1:attr.6">Sub_conjunctive</a> ) or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c3">S</font> is <a href="substut1.html#V7" title="SUBSTUT1:attr.7">Sub_universal</a> ) )
 </span><b>by </b><i><a class="ref" href="substut1.html#T12" target="_self" title="SUBSTUT1:th.12">Th12</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:74_1_2_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c4">y</font> = <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">S</font>;<br/><b>thus </b><a NAME="E2:74_1_2_1_1_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">S</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:74_1_2_1_1_1_2"/><i><font color="Green" title="E49">A6</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c3">S</font> is <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a> )
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c4">y</font> = <font color="Maroon">F<sub>2</sub></font>();<br/><b>thus </b><a NAME="E2:74_1_2_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">S</font>]
 <b>by </b><i><a class="txt" href="substut1.html#E1:74_1_2_1_1_1_2"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="substut1.html#T27" target="_self" title="SUBSTUT1:th.27">Th27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_1_1_1_3"><b>case </b></a><a NAME="E1:74_1_2_1_1_1_3"/><i><font color="Green" title="E49">A7</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c3">S</font> is <a href="substut1.html#V4" title="SUBSTUT1:attr.4">Sub_atomic</a> )
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c4">y</font> = <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c3">S</font>);<br/><b>thus </b><a NAME="E2:74_1_2_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">S</font>]
 <b>by </b><i><a class="txt" href="substut1.html#E1:74_1_2_1_1_1_3"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="substut1.html#T27" target="_self" title="SUBSTUT1:th.27">Th27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_1_1_1_4"><b>case </b></a><a NAME="E1:74_1_2_1_1_1_4"/><i><font color="Green" title="E49">A8</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c3">S</font> is <a href="substut1.html#V5" title="SUBSTUT1:attr.5">Sub_negative</a> )
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c4">y</font> = <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span>);<br/><b>thus </b><a NAME="E2:74_1_2_1_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">S</font>]
 <b>by </b><i><a class="txt" href="substut1.html#E1:74_1_2_1_1_1_4"><i><font color="Green" title="E49">A8</font></i></a>, <a class="ref" href="substut1.html#T27" target="_self" title="SUBSTUT1:th.27">Th27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_1_1_1_5"><b>case </b></a><a NAME="E1:74_1_2_1_1_1_5"/><i><font color="Green" title="E49">A9</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c3">S</font> is <a href="substut1.html#V6" title="SUBSTUT1:attr.6">Sub_conjunctive</a> )
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c4">y</font> = <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span>);<br/><b>thus </b><a NAME="E2:74_1_2_1_1_1_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">S</font>]
 <b>by </b><i><a class="txt" href="substut1.html#E1:74_1_2_1_1_1_5"><i><font color="Green" title="E49">A9</font></i></a>, <a class="ref" href="substut1.html#T27" target="_self" title="SUBSTUT1:th.27">Th27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_1_1_1_6"><b>case </b></a><a NAME="E1:74_1_2_1_1_1_6"/><i><font color="Green" title="E49">A10</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c3">S</font> is <a href="substut1.html#V7" title="SUBSTUT1:attr.7">Sub_universal</a> )
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c4">y</font> = <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c3">S</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</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> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span>);<br/><b>thus </b><a NAME="E2:74_1_2_1_1_1_6"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">S</font>]
 <b>by </b><i><a class="txt" href="substut1.html#E1:74_1_2_1_1_1_6"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="substut1.html#T27" target="_self" title="SUBSTUT1:th.27">Th27</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:74_1_2_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">S</font> ) &amp; (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">S</font>] ) )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E5:74_1_2"/><i><font color="Green" title="E49">A11</font></i>: 
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>4</sub></font>[<font color="Olive" title="b1">S</font>,<font color="Maroon" title="c3">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">S</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="substut1.html#E3:74_1_2"><i><font color="Green" title="E48">A5</font></i></a>);<br/></i>
<b>take </b><font color="Maroon" title="c4">H</font> = <font color="Maroon" title="c3">G</font>;<br/>

<b>thus </b><a NAME="E6:74_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">H</font>,<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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> ;<br/>

<b>assume </b><a NAME="E1:74_1_2_2"/><i><font color="Green" title="E50">A12</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/>

<b>thus </b><a NAME="E2:74_1_2_2"/>
( <font color="Maroon" title="c5">S</font> is <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a> implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_1_2_2_1_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:74_1_2_2_1_1_1_1"/><i><font color="Green" title="E51">A13</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:74_1_2_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E52">A14</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="substut1.html#E1:74_1_2_2"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/><a NAME="E3:74_1_2_2_1_1_1_1"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font>
 <b>by </b><i><a class="txt" href="substut1.html#E5:74_1_2"><i><font color="Green" title="E49">A11</font></i></a>, <a class="txt" href="substut1.html#E1:74_1_2_2_1_1_1_1"><i><font color="Green" title="E51">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E4:74_1_2_2_1_1_1_1"/>
( <font color="Maroon" title="c5">S</font> is <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a> implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 <b>by </b><i><a class="txt" href="substut1.html#E2:74_1_2"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="substut1.html#E2:74_1_2_2_1_1_1_1"><i><font color="Green" title="E52">A14</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:74_1_2_2_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:74_1_2_2_1_1_1_2"/>
( <font color="Maroon" title="c5">S</font> is <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a> implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 <b>by </b><i><a class="txt" href="substut1.html#E5:74_1_2"><i><font color="Green" title="E49">A11</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:74_1_2_2_1"/>
( <font color="Maroon" title="c5">S</font> is <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a> implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 ;<br/>


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

<b>thus </b><a NAME="E3:74_1_2_2"/>
( <font color="Maroon" title="c5">S</font> is <a href="substut1.html#V4" title="SUBSTUT1:attr.4">Sub_atomic</a> implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c5">S</font>) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_1_2_2_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:74_1_2_2_2_1_1_1"/><i><font color="Green" title="E51">A15</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:74_1_2_2_2_1_1_1"/><b>then </b><i><font color="Green" title="E52">A16</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="substut1.html#E1:74_1_2_2"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/><a NAME="E3:74_1_2_2_2_1_1_1"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font>
 <b>by </b><i><a class="txt" href="substut1.html#E5:74_1_2"><i><font color="Green" title="E49">A11</font></i></a>, <a class="txt" href="substut1.html#E1:74_1_2_2_2_1_1_1"><i><font color="Green" title="E51">A15</font></i></a></i>;<br/><b>hence </b><a NAME="E4:74_1_2_2_2_1_1_1"/>
( <font color="Maroon" title="c5">S</font> is <a href="substut1.html#V4" title="SUBSTUT1:attr.4">Sub_atomic</a> implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c5">S</font>) )
 <b>by </b><i><a class="txt" href="substut1.html#E2:74_1_2"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="substut1.html#E2:74_1_2_2_2_1_1_1"><i><font color="Green" title="E52">A16</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:74_1_2_2_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:74_1_2_2_2_1_1_2"/>
( <font color="Maroon" title="c5">S</font> is <a href="substut1.html#V4" title="SUBSTUT1:attr.4">Sub_atomic</a> implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c5">S</font>) )
 <b>by </b><i><a class="txt" href="substut1.html#E5:74_1_2"><i><font color="Green" title="E49">A11</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:74_1_2_2_2"/>
( <font color="Maroon" title="c5">S</font> is <a href="substut1.html#V4" title="SUBSTUT1:attr.4">Sub_atomic</a> implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c5">S</font>) )
 ;<br/>


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

<b>thus </b><a NAME="E4:74_1_2_2"/>
( <font color="Maroon" title="c5">S</font> is <a href="substut1.html#V5" title="SUBSTUT1:attr.5">Sub_negative</a> implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span></span>)</span>) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:74_1_2_2_3"/><i><font color="Green" title="E51">A17</font></i>: 
<font color="Maroon" title="c5">S</font> is <a href="substut1.html#V5" title="SUBSTUT1:attr.5">Sub_negative</a>
 ;<br/>

<a NAME="E2:74_1_2_2_3"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="substut1.html#E1:74_1_2_2"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="substut1.html#T22" target="_self" title="SUBSTUT1:th.22">Th22</a></i>;<br/>
<a NAME="E3:74_1_2_2_3"/><b>then </b><i><font color="Green" title="E52">A18</font></i>: 
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span>
 
<b>by </b><i><a class="txt" href="substut1.html#E5:74_1_2"><i><font color="Green" title="E49">A11</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_1_2_2_3_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:74_1_2_2_3_1_1_1"/><i><font color="Green" title="E53">A19</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:74_1_2_2_3_1_1_1"/><b>then </b><i><font color="Green" title="E54">A20</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="substut1.html#E1:74_1_2_2"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/><a NAME="E3:74_1_2_2_3_1_1_1"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font>
 <b>by </b><i><a class="txt" href="substut1.html#E5:74_1_2"><i><font color="Green" title="E49">A11</font></i></a>, <a class="txt" href="substut1.html#E1:74_1_2_2_3_1_1_1"><i><font color="Green" title="E53">A19</font></i></a></i>;<br/><b>hence </b><a NAME="E4:74_1_2_2_3_1_1_1"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span></span>)</span>)
 <b>by </b><i><a class="txt" href="substut1.html#E2:74_1_2"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="substut1.html#E1:74_1_2_2_3"><i><font color="Green" title="E51">A17</font></i></a>, <a class="txt" href="substut1.html#E3:74_1_2_2_3"><i><font color="Green" title="E52">A18</font></i></a>, <a class="txt" href="substut1.html#E2:74_1_2_2_3_1_1_1"><i><font color="Green" title="E54">A20</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:74_1_2_2_3_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:74_1_2_2_3_1_1_2"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span></span>)</span>)
 <b>by </b><i><a class="txt" href="substut1.html#E5:74_1_2"><i><font color="Green" title="E49">A11</font></i></a>, <a class="txt" href="substut1.html#E1:74_1_2_2_3"><i><font color="Green" title="E51">A17</font></i></a>, <a class="txt" href="substut1.html#E3:74_1_2_2_3"><i><font color="Green" title="E52">A18</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:74_1_2_2_3"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span></span>)</span>)
 ;<br/>


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

<b>thus </b><a NAME="E5:74_1_2_2"/>
( <font color="Maroon" title="c5">S</font> is <a href="substut1.html#V6" title="SUBSTUT1:attr.6">Sub_conjunctive</a> implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span></span>)</span>) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_2_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:74_1_2_2_4"/><i><font color="Green" title="E51">A21</font></i>: 
<font color="Maroon" title="c5">S</font> is <a href="substut1.html#V6" title="SUBSTUT1:attr.6">Sub_conjunctive</a>
 ;<br/>

<a NAME="E2:74_1_2_2_4"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="substut1.html#E1:74_1_2_2"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="substut1.html#T23" target="_self" title="SUBSTUT1:th.23">Th23</a></i>;<br/>
<a NAME="E3:74_1_2_2_4"/><b>then </b><i><font color="Green" title="E52">A22</font></i>: 
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span>
 
<b>by </b><i><a class="txt" href="substut1.html#E5:74_1_2"><i><font color="Green" title="E49">A11</font></i></a></i>;<br/>
<a NAME="E4:74_1_2_2_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="substut1.html#E1:74_1_2_2"><i><font color="Green" title="E50">A12</font></i></a>, <a class="txt" href="substut1.html#E1:74_1_2_2_4"><i><font color="Green" title="E51">A21</font></i></a>, <a class="ref" href="substut1.html#T23" target="_self" title="SUBSTUT1:th.23">Th23</a></i>;<br/>
<a NAME="E5:74_1_2_2_4"/><b>then </b><i><font color="Green" title="E53">A23</font></i>: 
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span>
 
<b>by </b><i><a class="txt" href="substut1.html#E5:74_1_2"><i><font color="Green" title="E49">A11</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_2_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_1_2_2_4_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_2_4_1_1_1"><b>suppose </b></a><a NAME="E1:74_1_2_2_4_1_1_1"/><i><font color="Green" title="E54">A24</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:74_1_2_2_4_1_1_1"/><b>then </b><i><font color="Green" title="E55">A25</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="substut1.html#E1:74_1_2_2"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/><a NAME="E3:74_1_2_2_4_1_1_1"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font>
 <b>by </b><i><a class="txt" href="substut1.html#E5:74_1_2"><i><font color="Green" title="E49">A11</font></i></a>, <a class="txt" href="substut1.html#E1:74_1_2_2_4_1_1_1"><i><font color="Green" title="E54">A24</font></i></a></i>;<br/><b>hence </b><a NAME="E4:74_1_2_2_4_1_1_1"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span></span>)</span>)
 <b>by </b><i><a class="txt" href="substut1.html#E2:74_1_2"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="substut1.html#E1:74_1_2_2_4"><i><font color="Green" title="E51">A21</font></i></a>, <a class="txt" href="substut1.html#E3:74_1_2_2_4"><i><font color="Green" title="E52">A22</font></i></a>, <a class="txt" href="substut1.html#E5:74_1_2_2_4"><i><font color="Green" title="E53">A23</font></i></a>, <a class="txt" href="substut1.html#E2:74_1_2_2_4_1_1_1"><i><font color="Green" title="E55">A25</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_2_4_1_1_2"><b>suppose </b></a><a NAME="E1:74_1_2_2_4_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:74_1_2_2_4_1_1_2"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span></span>)</span>)
 <b>by </b><i><a class="txt" href="substut1.html#E5:74_1_2"><i><font color="Green" title="E49">A11</font></i></a>, <a class="txt" href="substut1.html#E1:74_1_2_2_4"><i><font color="Green" title="E51">A21</font></i></a>, <a class="txt" href="substut1.html#E3:74_1_2_2_4"><i><font color="Green" title="E52">A22</font></i></a>, <a class="txt" href="substut1.html#E5:74_1_2_2_4"><i><font color="Green" title="E53">A23</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:74_1_2_2_4"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span></span>)</span>)
 ;<br/>


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

<b>thus </b><a NAME="E6:74_1_2_2"/>
( <font color="Maroon" title="c5">S</font> is <a href="substut1.html#V7" title="SUBSTUT1:attr.7">Sub_universal</a> implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c5">S</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</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> <font color="Maroon" title="c5">S</font></span>)</span></span>)</span>) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_2_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:74_1_2_2_5"/><i><font color="Green" title="E51">A26</font></i>: 
<font color="Maroon" title="c5">S</font> is <a href="substut1.html#V7" title="SUBSTUT1:attr.7">Sub_universal</a>
 ;<br/>

<a NAME="E2:74_1_2_2_5"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K31" title="SUBSTUT1:func.31">Sub_the_scope_of</a> <font color="Maroon" title="c5">S</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="substut1.html#E1:74_1_2_2"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="substut1.html#T24" target="_self" title="SUBSTUT1:th.24">Th24</a></i>;<br/>
<a NAME="E3:74_1_2_2_5"/><b>then </b><i><font color="Green" title="E52">A27</font></i>: 
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K31" title="SUBSTUT1:func.31">Sub_the_scope_of</a> <font color="Maroon" title="c5">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K31" title="SUBSTUT1:func.31">Sub_the_scope_of</a> <font color="Maroon" title="c5">S</font></span>)</span>
 
<b>by </b><i><a class="txt" href="substut1.html#E5:74_1_2"><i><font color="Green" title="E49">A11</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_2_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_2_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_1_2_2_5_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_2_5_1_1_1"><b>suppose </b></a><a NAME="E1:74_1_2_2_5_1_1_1"/><i><font color="Green" title="E53">A28</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:74_1_2_2_5_1_1_1"/><b>then </b><i><font color="Green" title="E54">A29</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="substut1.html#E1:74_1_2_2"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/><a NAME="E3:74_1_2_2_5_1_1_1"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font>
 <b>by </b><i><a class="txt" href="substut1.html#E5:74_1_2"><i><font color="Green" title="E49">A11</font></i></a>, <a class="txt" href="substut1.html#E1:74_1_2_2_5_1_1_1"><i><font color="Green" title="E53">A28</font></i></a></i>;<br/><b>hence </b><a NAME="E4:74_1_2_2_5_1_1_1"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c5">S</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</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> <font color="Maroon" title="c5">S</font></span>)</span></span>)</span>)
 <b>by </b><i><a class="txt" href="substut1.html#E2:74_1_2"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="substut1.html#E1:74_1_2_2_5"><i><font color="Green" title="E51">A26</font></i></a>, <a class="txt" href="substut1.html#E3:74_1_2_2_5"><i><font color="Green" title="E52">A27</font></i></a>, <a class="txt" href="substut1.html#E2:74_1_2_2_5_1_1_1"><i><font color="Green" title="E54">A29</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_2_5_1_1_2"><b>suppose </b></a><a NAME="E1:74_1_2_2_5_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:74_1_2_2_5_1_1_2"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c5">S</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</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> <font color="Maroon" title="c5">S</font></span>)</span></span>)</span>)
 <b>by </b><i><a class="txt" href="substut1.html#E5:74_1_2"><i><font color="Green" title="E49">A11</font></i></a>, <a class="txt" href="substut1.html#E1:74_1_2_2_5"><i><font color="Green" title="E51">A26</font></i></a>, <a class="txt" href="substut1.html#E3:74_1_2_2_5"><i><font color="Green" title="E52">A27</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:74_1_2_2_5"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c5">S</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</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> <font color="Maroon" title="c5">S</font></span>)</span></span>)</span>)
 ;<br/>


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


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


</div><b>end;</b></div>
<a NAME="E3:74_1"/><i><font color="Green" title="E47">A30</font></i>: 
for <font color="Olive" title="b1">n</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>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="substut1.html#E1:74_1"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="substut1.html#E2:74_1"><i><font color="Green" title="E46">A3</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <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 <br/>( <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; ( for <font color="Olive" title="b2">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">g</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>] holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">S</font> ) );<br/>
<a NAME="E4:74_1"/><i><font color="Green" title="E48">A31</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">y2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a>  &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y2</font>] holds <br/><font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y1</font>, <font color="Maroon" title="c3">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:74_1_3"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> 
 <b>and </b><br/><a NAME="E2:74_1_3"/><i><font color="Green" title="E49">A32</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y1</font>]
 <b>and </b><br/><a NAME="E3:74_1_3"/><i><font color="Green" title="E50">A33</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">y2</font>]
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">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> <b> such that </b><br/><a NAME="E5:74_1_3"/><i><font color="Green" title="E51">A34</font></i>: 
<font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 <b>and </b><br/><a NAME="E6:74_1_3"/><i><font color="Green" title="E52">A35</font></i>: 
for <font color="Olive" title="b1">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">g</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>] holds <br/><font color="Maroon" title="c2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">S</font>
 <b>by </b><i><a class="txt" href="substut1.html#E2:74_1_3"><i><font color="Green" title="E49">A32</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E8:74_1_3"/><i><font color="Green" title="E53">A36</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">F</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>]
 <b>by </b><i><a class="txt" href="substut1.html#E3:74_1"><i><font color="Green" title="E47">A30</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon" title="c2">y1</font> = 
<font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">S</font>
<b>by </b><i><a class="txt" href="substut1.html#E6:74_1_3"><i><font color="Green" title="E52">A35</font></i></a>, <a class="txt" href="substut1.html#E8:74_1_3"><i><font color="Green" title="E53">A36</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">y2</font>
<b>by </b><i><a class="txt" href="substut1.html#E3:74_1_3"><i><font color="Green" title="E50">A33</font></i></a>, <a class="txt" href="substut1.html#E5:74_1_3"><i><font color="Green" title="E51">A34</font></i></a>, <a class="txt" href="substut1.html#E8:74_1_3"><i><font color="Green" title="E53">A36</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:74_1"/><i><font color="Green" title="E49">A37</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a>  holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:74_1_4"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c2">x'</font> = <font color="Maroon" title="c1">x</font> as    <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/>
<b>consider </b><font color="Maroon" title="c3">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E4:74_1_4"/><i><font color="Green" title="E50">A38</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">F</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x'</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>]
 <b>by </b><i><a class="txt" href="substut1.html#E3:74_1"><i><font color="Green" title="E47">A30</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c2">x'</font>
;<br/>

<b>thus </b><a NAME="E5:74_1_4"/>
<font color="Maroon" title="c2">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">G</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E6:74_1_4"/><i><font color="Green" title="E51">A39</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">G</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x'</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>]
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>5</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> ] <b>means </b>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default">$1 <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x'</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> implies <font color="Maroon" title="c3">F</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" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 );<br/>
<div><i><font color="Green" title="E52">A40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">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> ;<br/><b>thus </b><a NAME="E1:74_1_4_1"/>
( <font color="Maroon" title="c5">S</font> is <a href="substut1.html#V4" title="SUBSTUT1:attr.4">Sub_atomic</a> implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">S</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_4_1_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:74_1_4_1_1"/><i><font color="Green" title="E53">A41</font></i>: 
<font color="Maroon" title="c5">S</font> is <a href="substut1.html#V4" title="SUBSTUT1:attr.4">Sub_atomic</a>
 <b>and </b><br/><a NAME="E2:74_1_4_1_1"/><i><font color="Green" title="E54">A42</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x'</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>
 ;<br/>

<b>thus </b><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> = 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c5">S</font>)
<b>by </b><i><a class="txt" href="substut1.html#E4:74_1_4"><i><font color="Green" title="E50">A38</font></i></a>, <a class="txt" href="substut1.html#E1:74_1_4_1_1"><i><font color="Green" title="E53">A41</font></i></a>, <a class="txt" href="substut1.html#E2:74_1_4_1_1"><i><font color="Green" title="E54">A42</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font>
<b>by </b><i><a class="txt" href="substut1.html#E6:74_1_4"><i><font color="Green" title="E51">A39</font></i></a>, <a class="txt" href="substut1.html#E1:74_1_4_1_1"><i><font color="Green" title="E53">A41</font></i></a>, <a class="txt" href="substut1.html#E2:74_1_4_1_1"><i><font color="Green" title="E54">A42</font></i></a></i>
;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:74_1_4_1"/>
( <font color="Maroon" title="c5">S</font> is <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a> implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">S</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_4_1_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:74_1_4_1_2"/><i><font color="Green" title="E53">A43</font></i>: 
<font color="Maroon" title="c5">S</font> is <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a>
 <b>and </b><br/><a NAME="E2:74_1_4_1_2"/><i><font color="Green" title="E54">A44</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x'</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>
 ;<br/>

<b>thus </b><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> = 
<font color="Maroon">F<sub>2</sub></font>()
<b>by </b><i><a class="txt" href="substut1.html#E4:74_1_4"><i><font color="Green" title="E50">A38</font></i></a>, <a class="txt" href="substut1.html#E1:74_1_4_1_2"><i><font color="Green" title="E53">A43</font></i></a>, <a class="txt" href="substut1.html#E2:74_1_4_1_2"><i><font color="Green" title="E54">A44</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font>
<b>by </b><i><a class="txt" href="substut1.html#E6:74_1_4"><i><font color="Green" title="E51">A39</font></i></a>, <a class="txt" href="substut1.html#E1:74_1_4_1_2"><i><font color="Green" title="E53">A43</font></i></a>, <a class="txt" href="substut1.html#E2:74_1_4_1_2"><i><font color="Green" title="E54">A44</font></i></a></i>
;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:74_1_4_1"/>
( <font color="Maroon" title="c5">S</font> is <a href="substut1.html#V5" title="SUBSTUT1:attr.5">Sub_negative</a> &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c5">S</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">S</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_4_1_3"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:74_1_4_1_3"/><i><font color="Green" title="E53">A45</font></i>: 
<font color="Maroon" title="c5">S</font> is <a href="substut1.html#V5" title="SUBSTUT1:attr.5">Sub_negative</a>
 <b>and </b><br/><a NAME="E2:74_1_4_1_3"/><i><font color="Green" title="E54">A46</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[ <a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c5">S</font>]
 <b>and </b><br/><a NAME="E3:74_1_4_1_3"/><i><font color="Green" title="E55">A47</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x'</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E4:74_1_4_1_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="substut1.html#E1:74_1_4_1_3"><i><font color="Green" title="E53">A45</font></i></a>, <a class="ref" href="substut1.html#T22" target="_self" title="SUBSTUT1:th.22">Th22</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> = 
<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span></span>)</span>)
<b>by </b><i><a class="txt" href="substut1.html#E4:74_1_4"><i><font color="Green" title="E50">A38</font></i></a>, <a class="txt" href="substut1.html#E1:74_1_4_1_3"><i><font color="Green" title="E53">A45</font></i></a>, <a class="txt" href="substut1.html#E2:74_1_4_1_3"><i><font color="Green" title="E54">A46</font></i></a>, <a class="txt" href="substut1.html#E3:74_1_4_1_3"><i><font color="Green" title="E55">A47</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
<br/>.= 
<font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font>
<b>by </b><i><a class="txt" href="substut1.html#E6:74_1_4"><i><font color="Green" title="E51">A39</font></i></a>, <a class="txt" href="substut1.html#E1:74_1_4_1_3"><i><font color="Green" title="E53">A45</font></i></a>, <a class="txt" href="substut1.html#E3:74_1_4_1_3"><i><font color="Green" title="E55">A47</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div><b>thus </b><a NAME="E4:74_1_4_1"/>
( <font color="Maroon" title="c5">S</font> is <a href="substut1.html#V6" title="SUBSTUT1:attr.6">Sub_conjunctive</a> &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Maroon" title="c5">S</font>] &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Maroon" title="c5">S</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">S</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_4_1_4"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:74_1_4_1_4"/><i><font color="Green" title="E53">A48</font></i>: 
<font color="Maroon" title="c5">S</font> is <a href="substut1.html#V6" title="SUBSTUT1:attr.6">Sub_conjunctive</a>
 <b>and </b><br/><a NAME="E2:74_1_4_1_4"/><i><font color="Green" title="E54">A49</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[ <a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Maroon" title="c5">S</font>]
 <b>and </b><br/><a NAME="E3:74_1_4_1_4"/><i><font color="Green" title="E55">A50</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[ <a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Maroon" title="c5">S</font>]
 <b>and </b><br/><a NAME="E4:74_1_4_1_4"/><i><font color="Green" title="E56">A51</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x'</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E5:74_1_4_1_4"/><i><font color="Green" title="E57">A52</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="substut1.html#E1:74_1_4_1_4"><i><font color="Green" title="E53">A48</font></i></a>, <a class="ref" href="substut1.html#T23" target="_self" title="SUBSTUT1:th.23">Th23</a></i>;<br/>
<a NAME="E6:74_1_4_1_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="substut1.html#E1:74_1_4_1_4"><i><font color="Green" title="E53">A48</font></i></a>, <a class="ref" href="substut1.html#T23" target="_self" title="SUBSTUT1:th.23">Th23</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> = 
<font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Maroon" title="c5">S</font></span>)</span></span>)</span>)
<b>by </b><i><a class="txt" href="substut1.html#E4:74_1_4"><i><font color="Green" title="E50">A38</font></i></a>, <a class="txt" href="substut1.html#E1:74_1_4_1_4"><i><font color="Green" title="E53">A48</font></i></a>, <a class="txt" href="substut1.html#E2:74_1_4_1_4"><i><font color="Green" title="E54">A49</font></i></a>, <a class="txt" href="substut1.html#E3:74_1_4_1_4"><i><font color="Green" title="E55">A50</font></i></a>, <a class="txt" href="substut1.html#E4:74_1_4_1_4"><i><font color="Green" title="E56">A51</font></i></a>, <a class="txt" href="substut1.html#E5:74_1_4_1_4"><i><font color="Green" title="E57">A52</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
<br/>.= 
<font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font>
<b>by </b><i><a class="txt" href="substut1.html#E6:74_1_4"><i><font color="Green" title="E51">A39</font></i></a>, <a class="txt" href="substut1.html#E1:74_1_4_1_4"><i><font color="Green" title="E53">A48</font></i></a>, <a class="txt" href="substut1.html#E4:74_1_4_1_4"><i><font color="Green" title="E56">A51</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div><b>thus </b><a NAME="E5:74_1_4_1"/>
( <font color="Maroon" title="c5">S</font> is <a href="substut1.html#V7" title="SUBSTUT1:attr.7">Sub_universal</a> &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="substut1.html#K31" title="SUBSTUT1:func.31">Sub_the_scope_of</a> <font color="Maroon" title="c5">S</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">S</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_4_1_5"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:74_1_4_1_5"/><i><font color="Green" title="E53">A53</font></i>: 
<font color="Maroon" title="c5">S</font> is <a href="substut1.html#V7" title="SUBSTUT1:attr.7">Sub_universal</a>
 <b>and </b><br/><a NAME="E2:74_1_4_1_5"/><i><font color="Green" title="E54">A54</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[ <a href="substut1.html#K31" title="SUBSTUT1:func.31">Sub_the_scope_of</a> <font color="Maroon" title="c5">S</font>]
 <b>and </b><br/><a NAME="E3:74_1_4_1_5"/><i><font color="Green" title="E55">A55</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x'</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E4:74_1_4_1_5"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K31" title="SUBSTUT1:func.31">Sub_the_scope_of</a> <font color="Maroon" title="c5">S</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="substut1.html#E1:74_1_4_1_5"><i><font color="Green" title="E53">A53</font></i></a>, <a class="ref" href="substut1.html#T24" target="_self" title="SUBSTUT1:th.24">Th24</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font> = 
<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c5">S</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</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> <font color="Maroon" title="c5">S</font></span>)</span></span>)</span>)
<b>by </b><i><a class="txt" href="substut1.html#E4:74_1_4"><i><font color="Green" title="E50">A38</font></i></a>, <a class="txt" href="substut1.html#E1:74_1_4_1_5"><i><font color="Green" title="E53">A53</font></i></a>, <a class="txt" href="substut1.html#E2:74_1_4_1_5"><i><font color="Green" title="E54">A54</font></i></a>, <a class="txt" href="substut1.html#E3:74_1_4_1_5"><i><font color="Green" title="E55">A55</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
<br/>.= 
<font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">S</font>
<b>by </b><i><a class="txt" href="substut1.html#E6:74_1_4"><i><font color="Green" title="E51">A39</font></i></a>, <a class="txt" href="substut1.html#E1:74_1_4_1_5"><i><font color="Green" title="E53">A53</font></i></a>, <a class="txt" href="substut1.html#E3:74_1_4_1_5"><i><font color="Green" title="E55">A55</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E8:74_1_4"/>
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>5</sub></font>[<font color="Olive" title="b1">S</font>]
 
<b>from </b><i><a class="ref" href="substut1.html#S2" target="_self" title="SUBSTUT1:sch.2">SUBSTUT1:sch 2</a>(<a class="txt" href="substut1.html#E7:74_1_4"><i><font color="Green" title="E52">A40</font></i></a>);<br/></i>
<b>hence </b><a NAME="E9:74_1_4"/>
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x'</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c1">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E7:74_1"/><i><font color="Green" title="E50">A56</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> 
 <b>and </b><br/><a NAME="E8:74_1"/><i><font color="Green" title="E51">A57</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a>  holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_1.html#S2" title="FUNCT_1:sch.2">FUNCT_1:sch 2</a>(<a class="txt" href="substut1.html#E4:74_1"><i><font color="Green" title="E48">A31</font></i></a>, <a class="txt" href="substut1.html#E5:74_1"><i><font color="Green" title="E49">A37</font></i></a>);<br/></i>
<a NAME="E9:74_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:74_1_5"/>
<font color="Maroon" title="c2">y</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="c1">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:74_1_5"/><i><font color="Green" title="E52">A58</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> 
 <b>and </b><br/><a NAME="E4:74_1_5"/><i><font color="Green" title="E53">A59</font></i>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="substut1.html#E7:74_1"><i><font color="Green" title="E50">A56</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">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> <b> such that </b><br/><a NAME="E6:74_1_5"/>
<font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <b>and </b><br/><a NAME="E7:74_1_5"/><i><font color="Green" title="E54">A60</font></i>: 
for <font color="Olive" title="b1">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">g</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>] holds <br/><font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">S</font>
 <b>by </b><i><a class="txt" href="substut1.html#E8:74_1"><i><font color="Green" title="E51">A57</font></i></a>, <a class="txt" href="substut1.html#E3:74_1_5"><i><font color="Green" title="E52">A58</font></i></a>, <a class="txt" href="substut1.html#E4:74_1_5"><i><font color="Green" title="E53">A59</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E9:74_1_5"/><i><font color="Green" title="E55">A61</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">G</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>]
 <b>by </b><i><a class="txt" href="substut1.html#E3:74_1"><i><font color="Green" title="E47">A30</font></i></a></i>;<br/>
<a NAME="E10:74_1_5"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">S</font>
 
<b>by </b><i><a class="txt" href="substut1.html#E7:74_1_5"><i><font color="Green" title="E54">A60</font></i></a>, <a class="txt" href="substut1.html#E9:74_1_5"><i><font color="Green" title="E55">A61</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:74_1_5"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c2">F</font> = <font color="Maroon" title="c1">F</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="substut1.html#E7:74_1"><i><font color="Green" title="E50">A56</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c2">F</font>
;<br/>

<b>let </b><font color="Maroon" title="c3">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> ;<br/><b>let </b><font color="Maroon" title="c4">d1</font>, <font color="Maroon" title="c5">d2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>



<b>consider </b><font color="Maroon" title="c6">S1</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> <b> such that </b><br/><a NAME="E12:74_1"/><i><font color="Green" title="E52">A62</font></i>: 
<font color="Maroon" title="c6">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S</font>
 <b>and </b><br/><a NAME="E13:74_1"/><i><font color="Green" title="E53">A63</font></i>: 
for <font color="Olive" title="b1">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">g</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">S1</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>] holds <br/><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">S1</font>
 <b>by </b><i><a class="txt" href="substut1.html#E8:74_1"><i><font color="Green" title="E51">A57</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E15:74_1"/><i><font color="Green" title="E54">A64</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">G</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">S1</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>]
 <b>by </b><i><a class="txt" href="substut1.html#E3:74_1"><i><font color="Green" title="E47">A30</font></i></a></i>;<br/>
<a NAME="E16:74_1"/><i><font color="Green" title="E55">A65</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="txt" href="substut1.html#E12:74_1"><i><font color="Green" title="E52">A62</font></i></a>, <a class="txt" href="substut1.html#E13:74_1"><i><font color="Green" title="E53">A63</font></i></a>, <a class="txt" href="substut1.html#E15:74_1"><i><font color="Green" title="E54">A64</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:74_1"/>
( <font color="Maroon" title="c3">S</font> is <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a> implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 <b>by </b><i><a class="txt" href="substut1.html#E12:74_1"><i><font color="Green" title="E52">A62</font></i></a>, <a class="txt" href="substut1.html#E15:74_1"><i><font color="Green" title="E54">A64</font></i></a></i>;<br/>

<b>thus </b><a NAME="E18:74_1"/>
( <font color="Maroon" title="c3">S</font> is <a href="substut1.html#V4" title="SUBSTUT1:attr.4">Sub_atomic</a> implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c3">S</font>) )
 <b>by </b><i><a class="txt" href="substut1.html#E12:74_1"><i><font color="Green" title="E52">A62</font></i></a>, <a class="txt" href="substut1.html#E15:74_1"><i><font color="Green" title="E54">A64</font></i></a>, <a class="txt" href="substut1.html#E16:74_1"><i><font color="Green" title="E55">A65</font></i></a></i>;<br/>

<a NAME="E19:74_1"/><i><font color="Green" title="E56">A66</font></i>: 
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>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">G</font>,<font color="Olive" title="b1">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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> ;<br/>

<b>assume </b><a NAME="E1:74_1_6"/><i><font color="Green" title="E57">A67</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c9">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> ;<br/>

<b>assume </b><a NAME="E2:74_1_6"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">S'</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 ;<br/>

<a NAME="E3:74_1_6"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">S'</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="substut1.html#E1:74_1_6"><i><font color="Green" title="E57">A67</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E4:74_1_6"/>
( ( <font color="Maroon" title="c9">S'</font> is <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a> implies <font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">S'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( <font color="Maroon" title="c9">S'</font> is <a href="substut1.html#V4" title="SUBSTUT1:attr.4">Sub_atomic</a> implies <font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">S'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c9">S'</font>) ) &amp; ( <font color="Maroon" title="c9">S'</font> is <a href="substut1.html#V5" title="SUBSTUT1:attr.5">Sub_negative</a> implies <font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">S'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c9">S'</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c9">S'</font> is <a href="substut1.html#V6" title="SUBSTUT1:attr.6">Sub_conjunctive</a> implies <font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">S'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Maroon" title="c9">S'</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Maroon" title="c9">S'</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c9">S'</font> is <a href="substut1.html#V7" title="SUBSTUT1:attr.7">Sub_universal</a> implies <font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">S'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c9">S'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</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> <font color="Maroon" title="c9">S'</font></span>)</span></span>)</span>) ) )
 <b>by </b><i><a class="txt" href="substut1.html#E12:74_1"><i><font color="Green" title="E52">A62</font></i></a>, <a class="txt" href="substut1.html#E15:74_1"><i><font color="Green" title="E54">A64</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E20:74_1"/>
( <font color="Maroon" title="c3">S</font> is <a href="substut1.html#V5" title="SUBSTUT1:attr.5">Sub_negative</a> &amp; <font color="Maroon" title="c4">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span> implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c4">d1</font>) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:74_1_7"/><i><font color="Green" title="E57">A68</font></i>: 
<font color="Maroon" title="c3">S</font> is <a href="substut1.html#V5" title="SUBSTUT1:attr.5">Sub_negative</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c8">S'</font> =  <a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c3">S</font>;<br/>
<b>set </b><font color="Maroon" title="c9">k</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>;<br/>
<a NAME="E2:74_1_7"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="substut1.html#E1:74_1_7"><i><font color="Green" title="E57">A68</font></i></a>, <a class="ref" href="substut1.html#T22" target="_self" title="SUBSTUT1:th.22">Th22</a></i>;<br/>
<a NAME="E3:74_1_7"/><b>then </b><i><font color="Green" title="E58">A69</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">G</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>]
 
<b>by </b><i><a class="txt" href="substut1.html#E19:74_1"><i><font color="Green" title="E56">A66</font></i></a></i>;<br/>
<a NAME="E4:74_1_7"/>
 ex <font color="Olive" title="b1">S1</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 <br/>( <font color="Olive" title="b1">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c3">S</font> &amp; ( for <font color="Olive" title="b2">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">g</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">S1</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>] holds <br/><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">S1</font> ) )
 
<b>by </b><i><a class="txt" href="substut1.html#E8:74_1"><i><font color="Green" title="E51">A57</font></i></a></i>;<br/>
<a NAME="E5:74_1_7"/><b>then </b>
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span>
 
<b>by </b><i><a class="txt" href="substut1.html#E3:74_1_7"><i><font color="Green" title="E58">A69</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:74_1_7"/>
(  not <font color="Maroon" title="c4">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span> or <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c4">d1</font>) )
 <b>by </b><i><a class="txt" href="substut1.html#E12:74_1"><i><font color="Green" title="E52">A62</font></i></a>, <a class="txt" href="substut1.html#E15:74_1"><i><font color="Green" title="E54">A64</font></i></a>, <a class="txt" href="substut1.html#E16:74_1"><i><font color="Green" title="E55">A65</font></i></a>, <a class="txt" href="substut1.html#E1:74_1_7"><i><font color="Green" title="E57">A68</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E21:74_1"/>
( <font color="Maroon" title="c3">S</font> is <a href="substut1.html#V6" title="SUBSTUT1:attr.6">Sub_conjunctive</a> &amp; <font color="Maroon" title="c4">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span> &amp; <font color="Maroon" title="c5">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span> implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">d2</font>) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:74_1_8"/><i><font color="Green" title="E57">A70</font></i>: 
<font color="Maroon" title="c3">S</font> is <a href="substut1.html#V6" title="SUBSTUT1:attr.6">Sub_conjunctive</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c8">S'</font> =  <a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Maroon" title="c3">S</font>;<br/>
<b>set </b><font color="Maroon" title="c9">k'</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">S''</font> =  <a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Maroon" title="c3">S</font>;<br/>
<b>set </b><font color="Maroon" title="c11">k''</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>;<br/>
<a NAME="E2:74_1_8"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="substut1.html#E1:74_1_8"><i><font color="Green" title="E57">A70</font></i></a>, <a class="ref" href="substut1.html#T23" target="_self" title="SUBSTUT1:th.23">Th23</a></i>;<br/>
<a NAME="E3:74_1_8"/><b>then </b><i><font color="Green" title="E58">A71</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">G</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>]
 
<b>by </b><i><a class="txt" href="substut1.html#E19:74_1"><i><font color="Green" title="E56">A66</font></i></a></i>;<br/>
<a NAME="E4:74_1_8"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="substut1.html#E1:74_1_8"><i><font color="Green" title="E57">A70</font></i></a>, <a class="ref" href="substut1.html#T23" target="_self" title="SUBSTUT1:th.23">Th23</a></i>;<br/>
<a NAME="E5:74_1_8"/><b>then </b><i><font color="Green" title="E59">A72</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">G</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>]
 
<b>by </b><i><a class="txt" href="substut1.html#E19:74_1"><i><font color="Green" title="E56">A66</font></i></a></i>;<br/>
<a NAME="E6:74_1_8"/><i><font color="Green" title="E60">A73</font></i>: 
 ex <font color="Olive" title="b1">S1</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 <br/>( <font color="Olive" title="b1">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Maroon" title="c3">S</font> &amp; ( for <font color="Olive" title="b2">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">g</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">S1</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>] holds <br/><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">S1</font> ) )
 
<b>by </b><i><a class="txt" href="substut1.html#E8:74_1"><i><font color="Green" title="E51">A57</font></i></a></i>;<br/>
<a NAME="E7:74_1_8"/><i><font color="Green" title="E61">A74</font></i>: 
 ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Maroon" title="c3">S</font> &amp; ( for <font color="Olive" title="b2">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">g</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">S2</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>] holds <br/><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">S2</font> ) )
 
<b>by </b><i><a class="txt" href="substut1.html#E8:74_1"><i><font color="Green" title="E51">A57</font></i></a></i>;<br/>
<a NAME="E8:74_1_8"/><i><font color="Green" title="E62">A75</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span>
 
<b>by </b><i><a class="txt" href="substut1.html#E3:74_1_8"><i><font color="Green" title="E58">A71</font></i></a>, <a class="txt" href="substut1.html#E6:74_1_8"><i><font color="Green" title="E60">A73</font></i></a></i>;<br/>
<a NAME="E9:74_1_8"/>
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span>
 
<b>by </b><i><a class="txt" href="substut1.html#E5:74_1_8"><i><font color="Green" title="E59">A72</font></i></a>, <a class="txt" href="substut1.html#E7:74_1_8"><i><font color="Green" title="E61">A74</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:74_1_8"/>
(  not <font color="Maroon" title="c4">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span> or  not <font color="Maroon" title="c5">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Maroon" title="c3">S</font></span>)</span> or <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">d2</font>) )
 <b>by </b><i><a class="txt" href="substut1.html#E12:74_1"><i><font color="Green" title="E52">A62</font></i></a>, <a class="txt" href="substut1.html#E15:74_1"><i><font color="Green" title="E54">A64</font></i></a>, <a class="txt" href="substut1.html#E16:74_1"><i><font color="Green" title="E55">A65</font></i></a>, <a class="txt" href="substut1.html#E1:74_1_8"><i><font color="Green" title="E57">A70</font></i></a>, <a class="txt" href="substut1.html#E8:74_1_8"><i><font color="Green" title="E62">A75</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E22:74_1"/><i><font color="Green" title="E57">A76</font></i>: 
<font color="Maroon" title="c3">S</font> is <a href="substut1.html#V7" title="SUBSTUT1:attr.7">Sub_universal</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c8">S'</font> =  <a href="substut1.html#K31" title="SUBSTUT1:func.31">Sub_the_scope_of</a> <font color="Maroon" title="c3">S</font>;<br/>
<b>set </b><font color="Maroon" title="c9">k</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K31" title="SUBSTUT1:func.31">Sub_the_scope_of</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>;<br/>
<a NAME="E23:74_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K31" title="SUBSTUT1:func.31">Sub_the_scope_of</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="substut1.html#E22:74_1"><i><font color="Green" title="E57">A76</font></i></a>, <a class="ref" href="substut1.html#T24" target="_self" title="SUBSTUT1:th.24">Th24</a></i>;<br/>
<a NAME="E24:74_1"/><b>then </b><i><font color="Green" title="E58">A77</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">G</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K31" title="SUBSTUT1:func.31">Sub_the_scope_of</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>]
 
<b>by </b><i><a class="txt" href="substut1.html#E19:74_1"><i><font color="Green" title="E56">A66</font></i></a></i>;<br/>
<a NAME="E25:74_1"/>
 ex <font color="Olive" title="b1">S1</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 <br/>( <font color="Olive" title="b1">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K31" title="SUBSTUT1:func.31">Sub_the_scope_of</a> <font color="Maroon" title="c3">S</font> &amp; ( for <font color="Olive" title="b2">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">g</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">S1</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>] holds <br/><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K31" title="SUBSTUT1:func.31">Sub_the_scope_of</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">S1</font> ) )
 
<b>by </b><i><a class="txt" href="substut1.html#E8:74_1"><i><font color="Green" title="E51">A57</font></i></a></i>;<br/>
<a NAME="E26:74_1"/><b>then </b>
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K31" title="SUBSTUT1:func.31">Sub_the_scope_of</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K31" title="SUBSTUT1:func.31">Sub_the_scope_of</a> <font color="Maroon" title="c3">S</font></span>)</span>
 
<b>by </b><i><a class="txt" href="substut1.html#E24:74_1"><i><font color="Green" title="E58">A77</font></i></a></i>;<br/>
<b>hence </b><a NAME="E27:74_1"/>
(  not <font color="Maroon" title="c4">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K31" title="SUBSTUT1:func.31">Sub_the_scope_of</a> <font color="Maroon" title="c3">S</font></span>)</span> or <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">d1</font>) )
 <b>by </b><i><a class="txt" href="substut1.html#E12:74_1"><i><font color="Green" title="E52">A62</font></i></a>, <a class="txt" href="substut1.html#E15:74_1"><i><font color="Green" title="E54">A64</font></i></a>, <a class="txt" href="substut1.html#E16:74_1"><i><font color="Green" title="E55">A65</font></i></a>, <a class="txt" href="substut1.html#E22:74_1"><i><font color="Green" title="E57">A76</font></i></a></i>;<br/>


</div>
