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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(),  <a href="funct_2.html#NM1">Function</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>(),   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ] <b>means </b>( ( <font color="Maroon">a<sub>3</sub></font> is <a href="substut1.html#V2" target="_self">Sub_VERUM</a> implies <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( <font color="Maroon">a<sub>3</sub></font> is <a href="substut1.html#V4" target="_self">Sub_atomic</a> implies <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a<sub>3</sub></font>) ) &amp; ( <font color="Maroon">a<sub>3</sub></font> is <a href="substut1.html#V5" target="_self">Sub_negative</a> implies <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">a<sub>3</sub></font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon">a<sub>3</sub></font> is <a href="substut1.html#V6" target="_self">Sub_conjunctive</a> implies <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">a<sub>3</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">a<sub>3</sub></font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon">a<sub>3</sub></font> is <a href="substut1.html#V7" target="_self">Sub_universal</a> implies <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">a<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Maroon">a<sub>3</sub></font></span>)</span></span>)</span>) ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="funct_2.html#NM1">Function</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>(),   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">S</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>   st  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Olive">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>2</sub></font> holds <br/>( ( <font color="Olive">S</font> is <a href="substut1.html#V2" target="_self">Sub_VERUM</a> implies <font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( <font color="Olive">S</font> is <a href="substut1.html#V4" target="_self">Sub_atomic</a> implies <font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">S</font>) ) &amp; ( <font color="Olive">S</font> is <a href="substut1.html#V5" target="_self">Sub_negative</a> implies <font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Olive">S</font></span>)</span></span>)</span>) ) &amp; ( <font color="Olive">S</font> is <a href="substut1.html#V6" target="_self">Sub_conjunctive</a> implies <font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Olive">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Olive">S</font></span>)</span></span>)</span>) ) &amp; ( <font color="Olive">S</font> is <a href="substut1.html#V7" target="_self">Sub_universal</a> implies <font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive">S</font>,<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Olive">S</font></span>)</span></span>)</span>) ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> ex <font color="Olive">F</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">F</font>,<font color="Maroon">a<sub>1</sub></font>];<br/>
<a NAME="E1:74_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<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">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>();<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>let </b><font color="Maroon">S</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ;<br/>

<b>assume </b><a NAME="E2:74_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<a NAME="E3:74_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#T34">QC_LANG1:34</a></i>;<br/>
<b>hence </b><a NAME="E4:74_1_1"/>
( ( <font color="Maroon">S</font> is <a href="substut1.html#V2" target="_self">Sub_VERUM</a> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( <font color="Maroon">S</font> is <a href="substut1.html#V4" target="_self">Sub_atomic</a> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">S</font>) ) &amp; ( <font color="Maroon">S</font> is <a href="substut1.html#V5" target="_self">Sub_negative</a> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">S</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon">S</font> is <a href="substut1.html#V6" target="_self">Sub_conjunctive</a> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">S</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon">S</font> is <a href="substut1.html#V7" target="_self">Sub_universal</a> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">S</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Maroon">S</font></span>)</span></span>)</span>) ) )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:74_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</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">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>given </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><a NAME="E1:74_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">F</font>,<font color="Maroon">n</font>]
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()] <b>means </b>( (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">F</font>,<font color="Maroon">a<sub>1</sub></font>] ) );<br/>
<a NAME="E2:74_1_2"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">S</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>   ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">S</font>,<font color="Olive">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">S</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">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">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 or (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">S</font> is <a href="substut1.html#V2" target="_self">Sub_VERUM</a> ) or (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">S</font> is <a href="substut1.html#V4" target="_self">Sub_atomic</a> ) or (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">S</font> is <a href="substut1.html#V5" target="_self">Sub_negative</a> ) or (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">S</font> is <a href="substut1.html#V6" target="_self">Sub_conjunctive</a> ) or (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">S</font> is <a href="substut1.html#V7" target="_self">Sub_universal</a> ) )
 </span><b>by </b><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">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>take </b><font color="Maroon">y</font> = <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font>;<br/><b>thus </b><a NAME="E2:74_1_2_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">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">E48</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">S</font> is <a href="substut1.html#V2" target="_self">Sub_VERUM</a> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">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">y</font>,<font color="Maroon">F</font>,<font color="Maroon">S</font>]
 <b>by </b><i>, </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">E49</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">S</font> is <a href="substut1.html#V4" target="_self">Sub_atomic</a> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">y</font> = <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">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">y</font>,<font color="Maroon">F</font>,<font color="Maroon">S</font>]
 <b>by </b><i><a class="txt" href="substut1.html#E1:74_1_2_1_1_1_3"><i><font color="Green">E49</font></i></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">E50</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">S</font> is <a href="substut1.html#V5" target="_self">Sub_negative</a> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">y</font> = <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">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">y</font>,<font color="Maroon">F</font>,<font color="Maroon">S</font>]
 <b>by </b><i>, </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">E51</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">S</font> is <a href="substut1.html#V6" target="_self">Sub_conjunctive</a> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">y</font> = <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">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">y</font>,<font color="Maroon">F</font>,<font color="Maroon">S</font>]
 <b>by </b><i>, </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">E52</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">S</font> is <a href="substut1.html#V7" target="_self">Sub_universal</a> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">y</font> = <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">S</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Maroon">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">y</font>,<font color="Maroon">F</font>,<font color="Maroon">S</font>]
 <b>by </b><i>, </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">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 implies <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> ) &amp; (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 implies <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Maroon">F</font>,<font color="Maroon">S</font>] ) )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">G</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E4:74_1_2"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">S</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">S</font>,<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">S</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>take </b><font color="Maroon">H</font> = <font color="Maroon">G</font>;<br/>

<b>thus </b><a NAME="E5:74_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">H</font>,<font color="Maroon">n</font> <a href="nat_1.html#K1">+</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">S</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ;<br/>

<b>assume </b><a NAME="E1:74_1_2_2"/><i><font color="Green">E75</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<b>thus </b><a NAME="E2:74_1_2_2"/>
( <font color="Maroon">S</font> is <a href="substut1.html#V2" target="_self">Sub_VERUM</a> implies <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</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">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 or  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</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">E76</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:74_1_2_2_1_1_1_1"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><a NAME="E3:74_1_2_2_1_1_1_1"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:74_1_2_2_1_1_1_1"/>
( <font color="Maroon">S</font> is <a href="substut1.html#V2" target="_self">Sub_VERUM</a> implies <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</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">E47</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">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:74_1_2_2_1_1_1_2"/>
( <font color="Maroon">S</font> is <a href="substut1.html#V2" target="_self">Sub_VERUM</a> implies <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 <b>by </b><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">S</font> is <a href="substut1.html#V2" target="_self">Sub_VERUM</a> implies <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</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">S</font> is <a href="substut1.html#V4" target="_self">Sub_atomic</a> implies <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">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">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 or  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</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">E78</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:74_1_2_2_2_1_1_1"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><a NAME="E3:74_1_2_2_2_1_1_1"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:74_1_2_2_2_1_1_1"/>
( <font color="Maroon">S</font> is <a href="substut1.html#V4" target="_self">Sub_atomic</a> implies <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">S</font>) )
 <b>by </b><i><a class="txt" href="substut1.html#E2:74_1_2"><i><font color="Green">E47</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">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:74_1_2_2_2_1_1_2"/>
( <font color="Maroon">S</font> is <a href="substut1.html#V4" target="_self">Sub_atomic</a> implies <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">S</font>) )
 <b>by </b><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">S</font> is <a href="substut1.html#V4" target="_self">Sub_atomic</a> implies <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">S</font>) )
 ;<br/>


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

<b>thus </b><a NAME="E4:74_1_2_2"/>
( <font color="Maroon">S</font> is <a href="substut1.html#V5" target="_self">Sub_negative</a> implies <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">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">E82</font></i>: 
<font color="Maroon">S</font> is <a href="substut1.html#V5" target="_self">Sub_negative</a>
 ;<br/>

<a NAME="E2:74_1_2_2_3"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="substut1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E3:74_1_2_2_3"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><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">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 or  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</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">E131</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:74_1_2_2_3_1_1_1"/><b>then </b><i><font color="Green">E132</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><a NAME="E3:74_1_2_2_3_1_1_1"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="txt" href="substut1.html#E4:74_1_2"><i><font color="Green">E53</font></i></a></i>;<br/><b>hence </b><a NAME="E4:74_1_2_2_3_1_1_1"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">S</font></span>)</span></span>)</span>)
 <b>by </b><i><a class="txt" href="substut1.html#E2:74_1_2"><i><font color="Green">E47</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">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:74_1_2_2_3_1_1_2"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">S</font></span>)</span></span>)</span>)
 <b>by </b><i>, , </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">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">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">S</font> is <a href="substut1.html#V6" target="_self">Sub_conjunctive</a> implies <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">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">E133</font></i>: 
<font color="Maroon">S</font> is <a href="substut1.html#V6" target="_self">Sub_conjunctive</a>
 ;<br/>

<a NAME="E2:74_1_2_2_4"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="txt" href="substut1.html#E1:74_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E3:74_1_2_2_4"/><b>then </b><i><font color="Green">E134</font></i>: 
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:74_1_2_2_4"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, , <a class="txt" href="substut1.html#E1:74_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E5:74_1_2_2_4"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><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">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 or  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</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">E136</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:74_1_2_2_4_1_1_1"/><b>then </b><i><font color="Green">E137</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><a NAME="E3:74_1_2_2_4_1_1_1"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:74_1_2_2_4_1_1_1"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">S</font></span>)</span></span>)</span>)
 <b>by </b><i><a class="txt" href="substut1.html#E2:74_1_2"><i><font color="Green">E47</font></i></a>, , , <a class="txt" href="substut1.html#E4:74_1_2"><i><font color="Green">E53</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">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:74_1_2_2_4_1_1_2"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">S</font></span>)</span></span>)</span>)
 <b>by </b><i>, , , <a class="txt" href="substut1.html#E4:74_1_2"><i><font color="Green">E53</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">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">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">S</font> is <a href="substut1.html#V7" target="_self">Sub_universal</a> implies <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">S</font>,<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Maroon">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">E138</font></i>: 
<font color="Maroon">S</font> is <a href="substut1.html#V7" target="_self">Sub_universal</a>
 ;<br/>

<a NAME="E2:74_1_2_2_5"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Maroon">S</font></span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:74_1_2_2_5"/><b>then </b><i><font color="Green">E139</font></i>: 
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><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">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 or  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</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">E140</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:74_1_2_2_5_1_1_1"/><b>then </b><i><font color="Green">E141</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><a NAME="E3:74_1_2_2_5_1_1_1"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="txt" href="substut1.html#E4:74_1_2"><i><font color="Green">E53</font></i></a></i>;<br/><b>hence </b><a NAME="E4:74_1_2_2_5_1_1_1"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">S</font>,<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Maroon">S</font></span>)</span></span>)</span>)
 <b>by </b><i><a class="txt" href="substut1.html#E2:74_1_2"><i><font color="Green">E47</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">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:74_1_2_2_5_1_1_2"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">S</font>,<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Maroon">S</font></span>)</span></span>)</span>)
 <b>by </b><i>, , </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">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">S</font>,<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Maroon">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">E142</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="substut1.html#E2:74_1"><i><font color="Green">E45</font></i></a>, );<br/></i>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">S</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>  st <br/>( <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Olive">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>] holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">S</font> ) );<br/>
<a NAME="E4:74_1"/><i><font color="Green">E143</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>  &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">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">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:74_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> 
 <b>and </b><br/><a NAME="E2:74_1_3"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y1</font>]
 <b>and </b><br/><a NAME="E3:74_1_3"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y2</font>]
 ;<br/>

<b>consider </b><font color="Maroon">S</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> <b> such that </b><br/><a NAME="E5:74_1_3"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E6:74_1_3"/><i><font color="Green">E147</font></i>: 
for <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>] holds <br/><font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="substut1.html#K16" target="_self">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">E148</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">F</font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>]
 <b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">y1</font> = 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">y2</font>
<b>by </b><i>, , </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:74_1"/><i><font color="Green">E149</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>  holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>,<font color="Olive">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:74_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>  ;<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="substut1.html#K16" target="_self">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">E150</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">F</font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>]
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
;<br/>

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

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

<b>let </b><font color="Maroon">G</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="substut1.html#K16" target="_self">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">E151</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">G</font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>]
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ] <b>means </b>(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<div><i><font color="Green">E199</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">S</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ;<br/><b>thus </b><a NAME="E1:74_1_4_1"/>
( <font color="Maroon">S</font> is <a href="substut1.html#V4" target="_self">Sub_atomic</a> implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">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">E200</font></i>: 
<font color="Maroon">S</font> is <a href="substut1.html#V4" target="_self">Sub_atomic</a>
 <b>and </b><br/><a NAME="E2:74_1_4_1_1"/><i><font color="Green">E201</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>
 ;<br/>

<b>thus </b><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> = 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">S</font>)
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font>
<b>by </b><i>, , </i>
;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:74_1_4_1"/>
( <font color="Maroon">S</font> is <a href="substut1.html#V2" target="_self">Sub_VERUM</a> implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">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">E202</font></i>: 
<font color="Maroon">S</font> is <a href="substut1.html#V2" target="_self">Sub_VERUM</a>
 <b>and </b><br/><a NAME="E2:74_1_4_1_2"/><i><font color="Green">E203</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>
 ;<br/>

<b>thus </b><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> = 
<font color="Maroon">F<sub>2</sub></font>()
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font>
<b>by </b><i>, , </i>
;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:74_1_4_1"/>
( <font color="Maroon">S</font> is <a href="substut1.html#V5" target="_self">Sub_negative</a> &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">S</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">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">E204</font></i>: 
<font color="Maroon">S</font> is <a href="substut1.html#V5" target="_self">Sub_negative</a>
 <b>and </b><br/><a NAME="E2:74_1_4_1_3"/><i><font color="Green">E205</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[ <a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">S</font>]
 <b>and </b><br/><a NAME="E3:74_1_4_1_3"/><i><font color="Green">E206</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E4:74_1_4_1_3"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="substut1.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> = 
<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">S</font></span>)</span></span>)</span>)
<b>by </b><i>, , , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font>
<b>by </b><i>, , </i>
;<br/><br/>


</div><b>end;</b></div><b>thus </b><a NAME="E4:74_1_4_1"/>
( <font color="Maroon">S</font> is <a href="substut1.html#V6" target="_self">Sub_conjunctive</a> &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font>] &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">S</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">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">E207</font></i>: 
<font color="Maroon">S</font> is <a href="substut1.html#V6" target="_self">Sub_conjunctive</a>
 <b>and </b><br/><a NAME="E2:74_1_4_1_4"/><i><font color="Green">E208</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[ <a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font>]
 <b>and </b><br/><a NAME="E3:74_1_4_1_4"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[ <a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">S</font>]
 <b>and </b><br/><a NAME="E4:74_1_4_1_4"/><i><font color="Green">E210</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E5:74_1_4_1_4"/><i><font color="Green">E211</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="substut1.html#E1:74_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E6:74_1_4_1_4"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="substut1.html#E1:74_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>hence </b><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> = 
<font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">S</font></span>)</span></span>)</span>)
<b>by </b><i>, , , , , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font>
<b>by </b><i>, , </i>
;<br/><br/>


</div><b>end;</b></div><b>thus </b><a NAME="E5:74_1_4_1"/>
( <font color="Maroon">S</font> is <a href="substut1.html#V7" target="_self">Sub_universal</a> &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Maroon">S</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">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">E212</font></i>: 
<font color="Maroon">S</font> is <a href="substut1.html#V7" target="_self">Sub_universal</a>
 <b>and </b><br/><a NAME="E2:74_1_4_1_5"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[ <a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Maroon">S</font>]
 <b>and </b><br/><a NAME="E3:74_1_4_1_5"/><i><font color="Green">E214</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E4:74_1_4_1_5"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Maroon">S</font></span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> = 
<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">S</font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Maroon">S</font></span>)</span></span>)</span>)
<b>by </b><i>, , , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font>
<b>by </b><i>, , </i>
;<br/><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E8:74_1_4"/>
for <font color="Olive">S</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>  holds  <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">S</font>]
 
<b>from </b><i><a class="ref" href="substut1.html#S2" target="_self">SUBSTUT1:sch 2</a>();<br/></i>
<b>hence </b><a NAME="E9:74_1_4"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:74_1"/><i><font color="Green">E215</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> 
 <b>and </b><br/><a NAME="E8:74_1"/><i><font color="Green">E216</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>  holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>,<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<a NAME="E9:74_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">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">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:74_1_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:74_1_5"/><i><font color="Green">E217</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> 
 <b>and </b><br/><a NAME="E4:74_1_5"/><i><font color="Green">E218</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">S</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> <b> such that </b><br/><a NAME="E6:74_1_5"/>
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E7:74_1_5"/><i><font color="Green">E219</font></i>: 
for <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>] holds <br/><font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font>
 <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">G</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="substut1.html#K16" target="_self">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">E220</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">G</font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>]
 <b>by </b><i/>;<br/>
<a NAME="E10:74_1_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E11:74_1_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>() <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>let </b><font color="Maroon">S</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ;<br/><b>let </b><font color="Maroon">d1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">d2</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>



<b>consider </b><font color="Maroon">S1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> <b> such that </b><br/><a NAME="E12:74_1"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E13:74_1"/><i><font color="Green">E222</font></i>: 
for <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>] holds <br/><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S1</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">G</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="substut1.html#K16" target="_self">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">E223</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">G</font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>]
 <b>by </b><i/>;<br/>
<a NAME="E16:74_1"/><i><font color="Green">E224</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E17:74_1"/>
( <font color="Maroon">S</font> is <a href="substut1.html#V2" target="_self">Sub_VERUM</a> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E18:74_1"/>
( <font color="Maroon">S</font> is <a href="substut1.html#V4" target="_self">Sub_atomic</a> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">S</font>) )
 <b>by </b><i>, , </i>;<br/>

<a NAME="E19:74_1"/><i><font color="Green">E225</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">G</font>,<font color="Olive">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">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:74_1_6"/><i><font color="Green">E226</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">S'</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ;<br/>

<b>assume </b><a NAME="E2:74_1_6"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S'</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E3:74_1_6"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S'</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>hence </b><a NAME="E4:74_1_6"/>
( ( <font color="Maroon">S'</font> is <a href="substut1.html#V2" target="_self">Sub_VERUM</a> implies <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( <font color="Maroon">S'</font> is <a href="substut1.html#V4" target="_self">Sub_atomic</a> implies <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">S'</font>) ) &amp; ( <font color="Maroon">S'</font> is <a href="substut1.html#V5" target="_self">Sub_negative</a> implies <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">S'</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon">S'</font> is <a href="substut1.html#V6" target="_self">Sub_conjunctive</a> implies <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S'</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">S'</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon">S'</font> is <a href="substut1.html#V7" target="_self">Sub_universal</a> implies <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">S'</font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Maroon">S'</font></span>)</span></span>)</span>) ) )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E20:74_1"/>
( <font color="Maroon">S</font> is <a href="substut1.html#V5" target="_self">Sub_negative</a> &amp; <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">S</font></span>)</span> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">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">E227</font></i>: 
<font color="Maroon">S</font> is <a href="substut1.html#V5" target="_self">Sub_negative</a>
 ;<br/>

<b>set </b><font color="Maroon">S'</font> =  <a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">S</font>;<br/>
<b>set </b><font color="Maroon">k</font> =  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>;<br/>
<a NAME="E2:74_1_7"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="substut1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E3:74_1_7"/><b>then </b><i><font color="Green">E228</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">G</font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>]
 
<b>by </b><i/>;<br/>
<a NAME="E4:74_1_7"/>
 ex <font color="Olive">S1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>  st <br/>( <font color="Olive">S1</font> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">S</font> &amp; ( for <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Olive">S1</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>] holds <br/><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">S1</font> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E5:74_1_7"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:74_1_7"/>
(  not <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <font color="Maroon">S</font></span>)</span> or <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">d1</font>) )
 <b>by </b><i>, , , </i>;<br/>


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

<b>thus </b><a NAME="E21:74_1"/>
( <font color="Maroon">S</font> is <a href="substut1.html#V6" target="_self">Sub_conjunctive</a> &amp; <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font></span>)</span> &amp; <font color="Maroon">d2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">S</font></span>)</span> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">d1</font>,<font color="Maroon">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">E229</font></i>: 
<font color="Maroon">S</font> is <a href="substut1.html#V6" target="_self">Sub_conjunctive</a>
 ;<br/>

<b>set </b><font color="Maroon">S'</font> =  <a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font>;<br/>
<b>set </b><font color="Maroon">k'</font> =  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">S''</font> =  <a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">S</font>;<br/>
<b>set </b><font color="Maroon">k''</font> =  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>;<br/>
<a NAME="E2:74_1_8"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="substut1.html#E1:74_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E3:74_1_8"/><b>then </b><i><font color="Green">E233</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">G</font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>]
 
<b>by </b><i/>;<br/>
<a NAME="E4:74_1_8"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="substut1.html#E1:74_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E5:74_1_8"/><b>then </b><i><font color="Green">E234</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">G</font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>]
 
<b>by </b><i/>;<br/>
<a NAME="E6:74_1_8"/><i><font color="Green">E235</font></i>: 
 ex <font color="Olive">S1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>  st <br/>( <font color="Olive">S1</font> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font> &amp; ( for <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Olive">S1</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>] holds <br/><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">S1</font> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E7:74_1_8"/><i><font color="Green">E236</font></i>: 
 ex <font color="Olive">S2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>  st <br/>( <font color="Olive">S2</font> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">S</font> &amp; ( for <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Olive">S2</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>] holds <br/><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">S2</font> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E8:74_1_8"/><i><font color="Green">E237</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:74_1_8"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="substut1.html#D8" target="_self">Def8</a>, </i>;<br/>
<b>hence </b><a NAME="E10:74_1_8"/>
(  not <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font></span>)</span> or  not <font color="Maroon">d2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">S</font></span>)</span> or <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">d1</font>,<font color="Maroon">d2</font>) )
 <b>by </b><i>, , , , </i>;<br/>


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

<b>assume </b><a NAME="E22:74_1"/><i><font color="Green">E238</font></i>: 
<font color="Maroon">S</font> is <a href="substut1.html#V7" target="_self">Sub_universal</a>
 ;<br/>

<b>set </b><font color="Maroon">S'</font> =  <a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Maroon">S</font>;<br/>
<b>set </b><font color="Maroon">k</font> =  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Maroon">S</font></span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>;<br/>
<a NAME="E23:74_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Maroon">S</font></span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E24:74_1"/><b>then </b><i><font color="Green">E239</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">G</font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Maroon">S</font></span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>]
 
<b>by </b><i/>;<br/>
<a NAME="E25:74_1"/>
 ex <font color="Olive">S1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>  st <br/>( <font color="Olive">S1</font> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Maroon">S</font> &amp; ( for <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Olive">S1</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>] holds <br/><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">S1</font> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E26:74_1"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E27:74_1"/>
(  not <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <font color="Maroon">S</font></span>)</span> or <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">S</font>,<font color="Maroon">d1</font>) )
 <b>by </b><i>, , , </i>;<br/>


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