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

<b>let </b><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">ll</font> be   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</font>;<br/><b>let </b><font color="Maroon" title="c4">v</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c2">A</font>;<br/><b>let </b><font color="Maroon" title="c5">vS</font>, <font color="Maroon" title="c6">vS1</font>, <font color="Maroon" title="c7">vS2</font> be   <a href="sublemma.html#NM1" title="SUBLEMMA:NM.1">Val_Sub</a> of <font color="Maroon" title="c2">A</font>;<br/>













<b>assume </b><a NAME="E1:91"/><i><font color="Green" title="E74">A1</font></i>: 
( ( for <font color="Olive" title="b1">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">vS1</font> holds <br/> not <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">still_not-bound_in</a> <font color="Maroon" title="c3">ll</font> ) &amp; ( for <font color="Olive" title="b1">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS2</font> holds <br/><font color="Maroon" title="c7">vS2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> ) &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">vS</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS2</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c8">ll1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c5">vS</font></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font>;<br/>
<b>set </b><font color="Maroon" title="c9">ll2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font>;<br/>
<a NAME="E2:91"/><i><font color="Green" title="E75">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c5">vS</font></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 
<b>by </b><i><a class="ref" href="valuat_1.html#D8" title="VALUAT_1:def.8">VALUAT_1:def 8</a></i>;<br/>
<a NAME="E3:91"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">k</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c5">vS</font></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:91_1"/><i><font color="Green" title="E76">A3</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">k</font>
 ;<br/>

<a NAME="E2:91_1"/><i><font color="Green" title="E77">A4</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sublemma.html#E2:91"><i><font color="Green" title="E75">A2</font></i></a>, <a class="txt" href="sublemma.html#E1:91_1"><i><font color="Green" title="E76">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:91_1"/><i><font color="Green" title="E78">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">ll</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T44" title="RELAT_1:th.44">RELAT_1:44</a></i>;<br/>
<div><i><font color="Green" title="E79">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:91_1_1"/><i><font color="Green" title="E80">A7</font></i>: 
not <font color="Maroon" title="c3">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span>
 ;<br/><a NAME="E2:91_1_1"/><b>then </b>
not <font color="Maroon" title="c3">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></i>;<br/><a NAME="E3:91_1_1"/><b>then </b>
not <font color="Maroon" title="c3">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">vS</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></i>;<br/><a NAME="E4:91_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c5">vS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>;<br/><a NAME="E5:91_1_1"/><b>then </b><i><font color="Green" title="E81">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c5">vS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 ;<br/><a NAME="E6:91_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c5">vS</font></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E2:91"><i><font color="Green" title="E75">A2</font></i></a>, <a class="txt" href="sublemma.html#E1:91_1"><i><font color="Green" title="E76">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E7:91_1_1"/><b>then </b><i><font color="Green" title="E82">A9</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c5">vS</font></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E5:91_1_1"><i><font color="Green" title="E81">A8</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><a NAME="E8:91_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E1:91_1_1"><i><font color="Green" title="E80">A7</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>;<br/><a NAME="E9:91_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 ;<br/><b>hence </b><a NAME="E10:91_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c5">vS</font></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E2:91_1"><i><font color="Green" title="E77">A4</font></i></a>, <a class="txt" href="sublemma.html#E7:91_1_1"><i><font color="Green" title="E82">A9</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:91_1_2"/><i><font color="Green" title="E80">A10</font></i>: 
<font color="Maroon" title="c3">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon" title="c11">x</font> = <font color="Maroon" title="c3">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> as   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> <b>by </b><i><a class="txt" href="sublemma.html#E2:91_1"><i><font color="Green" title="E77">A4</font></i></a>, <a class="txt" href="sublemma.html#E3:91_1"><i><font color="Green" title="E78">A5</font></i></a>, <a class="ref" href="sublemma.html#T5" target="_self" title="SUBLEMMA:th.5">Th5</a></i>;<br/><div><i><font color="Green" title="E81">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:91_1_2_1"/><i><font color="Green" title="E82">A12</font></i>: 
not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS2</font>
 ;<br/><a NAME="E2:91_1_2_1"/><b>then </b><i><font color="Green" title="E83">A13</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E1:91_1_2"><i><font color="Green" title="E80">A10</font></i></a>, <a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></i>;<br/><a NAME="E3:91_1_2_1"/><i><font color="Green" title="E84">A14</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E1:91_1_2_1"><i><font color="Green" title="E82">A12</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>;<br/><div><i><font color="Green" title="E85">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:91_1_2_1_1"/><i><font color="Green" title="E86">A16</font></i>: 
not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">vS1</font>
 ;<br/><a NAME="E2:91_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E3:91_1_2_1"><i><font color="Green" title="E84">A14</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>;<br/><a NAME="E3:91_1_2_1_1"/><b>then </b><i><font color="Green" title="E87">A17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E1:91_1_2"><i><font color="Green" title="E80">A10</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/><a NAME="E4:91_1_2_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">vS</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E2:91_1_2_1"><i><font color="Green" title="E83">A13</font></i></a>, <a class="txt" href="sublemma.html#E1:91_1_2_1_1"><i><font color="Green" title="E86">A16</font></i></a>, <a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></i>;<br/><a NAME="E5:91_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c5">vS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E3:91_1_2_1_1"><i><font color="Green" title="E87">A17</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/><a NAME="E6:91_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c5">vS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
 ;<br/><a NAME="E7:91_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c5">vS</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font>
 ;<br/><a NAME="E8:91_1_2_1_1"/><b>then </b><i><font color="Green" title="E88">A18</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c5">vS</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E2:91_1"><i><font color="Green" title="E77">A4</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><a NAME="E9:91_1_2_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c5">vS</font></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E2:91"><i><font color="Green" title="E75">A2</font></i></a>, <a class="txt" href="sublemma.html#E1:91_1"><i><font color="Green" title="E76">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E10:91_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c5">vS</font></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E8:91_1_2_1_1"><i><font color="Green" title="E88">A18</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:91_1_2_1_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">vS1</font>
 ;<br/><a NAME="E2:91_1_2_1_2"/><b>then </b><i><font color="Green" title="E86">A19</font></i>: 
not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">still_not-bound_in</a> <font color="Maroon" title="c3">ll</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E1:91"><i><font color="Green" title="E74">A1</font></i></a></i>;<br/><a NAME="E3:91_1_2_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">ll</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="ref" href="substut1.html#T34" title="SUBSTUT1:th.34">SUBSTUT1:34</a></i>;<br/><a NAME="E4:91_1_2_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">ll</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  )
 <b>by </b><i><a class="txt" href="sublemma.html#E1:91_1"><i><font color="Green" title="E76">A3</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:91_1_2_1_2"/><i><font color="Green" title="E87">A20</font></i>: 
( <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font> &amp; <font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">ll</font> &amp; <font color="Maroon" title="c3">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c3">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  )
 ;<br/><a NAME="E7:91_1_2_1_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">ll</font> &amp; <font color="Maroon" title="c3">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="sublemma.html#E6:91_1_2_1_2"><i><font color="Green" title="E87">A20</font></i></a></i>;<br/><a NAME="E8:91_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang3.html#K1" title="QC_LANG3:func.1">variables_in</a> <font color="Maroon" title="c3">ll</font>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> 
 <b>by </b><i><a class="ref" href="qc_lang3.html#D2" title="QC_LANG3:def.2">QC_LANG3:def 2</a></i>;<br/><b>hence </b><a NAME="E9:91_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="sublemma.html#E2:91_1_2_1_2"><i><font color="Green" title="E86">A19</font></i></a>, <a class="ref" href="qc_lang3.html#T6" title="QC_LANG3:th.6">QC_LANG3:6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:91_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c5">vS</font></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E4:91_1_2_1"><i><font color="Green" title="E85">A15</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:91_1_2_2"/><i><font color="Green" title="E82">A21</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS2</font>
 ;<br/><a NAME="E2:91_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">vS2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/><a NAME="E3:91_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E1:91"><i><font color="Green" title="E74">A1</font></i></a>, <a class="txt" href="sublemma.html#E1:91_1_2_2"><i><font color="Green" title="E82">A21</font></i></a></i>;<br/><a NAME="E4:91_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E1:91_1_2"><i><font color="Green" title="E80">A10</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/><a NAME="E5:91_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font>
 ;<br/><a NAME="E6:91_1_2_2"/><b>then </b><i><font color="Green" title="E83">A22</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E2:91_1"><i><font color="Green" title="E77">A4</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><a NAME="E7:91_1_2_2"/>
not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">vS</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E1:91"><i><font color="Green" title="E74">A1</font></i></a>, <a class="txt" href="sublemma.html#E1:91_1_2_2"><i><font color="Green" title="E82">A21</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E8:91_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c5">vS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>;<br/><a NAME="E9:91_1_2_2"/><b>then </b><i><font color="Green" title="E84">A23</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c5">vS</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font>
 ;<br/><a NAME="E10:91_1_2_2"/>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c5">vS</font></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E2:91"><i><font color="Green" title="E75">A2</font></i></a>, <a class="txt" href="sublemma.html#E1:91_1"><i><font color="Green" title="E76">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E11:91_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c5">vS</font></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E6:91_1_2_2"><i><font color="Green" title="E83">A22</font></i></a>, <a class="txt" href="sublemma.html#E9:91_1_2_2"><i><font color="Green" title="E84">A23</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:91_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c5">vS</font></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E3:91_1_2"><i><font color="Green" title="E81">A11</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:91_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c5">vS</font></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E4:91_1"><i><font color="Green" title="E79">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:91"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c5">vS</font></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS2</font></span>)</span></span>)</span> <a href="valuat_1.html#K8" title="VALUAT_1:func.8">*'</a> <font color="Maroon" title="c3">ll</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E2:91"><i><font color="Green" title="E75">A2</font></i></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></i>;<br/>


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