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

<b>let </b><font color="Maroon" title="c1">p</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/> for <font color="Olive" title="b2">Sub</font> being  <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> holds   <a href="substut1.html#K14" title="SUBSTUT1:func.14">ExpandSub</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Olive" title="b1">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Olive" title="b2">Sub</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Olive" title="b1">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Olive" title="b2">Sub</font></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3"><a href="substut2.html#K1" title="SUBSTUT2:func.1">[</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Olive" title="b2">Sub</font></span><a href="substut2.html#K1" title="SUBSTUT2:func.1">]</a></span></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Sub</font> being  <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> holds   <a href="substut1.html#K14" title="SUBSTUT1:func.14">ExpandSub</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Olive" title="b1">Sub</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Olive" title="b1">Sub</font></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3"><a href="substut2.html#K1" title="SUBSTUT2:func.1">[</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Olive" title="b1">Sub</font></span><a href="substut2.html#K1" title="SUBSTUT2:func.1">]</a></span></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">Sub</font> be   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="substut1.html#K14" title="SUBSTUT1:func.14">ExpandSub</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3"><a href="substut2.html#K1" title="SUBSTUT2:func.1">[</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span><a href="substut2.html#K1" title="SUBSTUT2:func.1">]</a></span></span>)</span></span>)</span></span><br/>





<b>set </b><font color="Maroon" title="c4">finSub</font> =  <a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font>;<br/>
<b>set </b><font color="Maroon" title="c5">z</font> =  <a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p1"><a href="substut2.html#K1" title="SUBSTUT2:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span><a href="substut2.html#K1" title="SUBSTUT2:func.1">]</a></span>;<br/>
<div><i><font color="Green" title="E10">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1"/><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="substut1.html#K14" title="SUBSTUT1:func.14">ExpandSub</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3"><a href="substut2.html#K1" title="SUBSTUT2:func.1">[</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span><a href="substut2.html#K1" title="SUBSTUT2:func.1">]</a></span></span>)</span></span>)</span></span><br/><a NAME="E2:10_1"/><i><font color="Green" title="E12">A3</font></i>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<span class="p3">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <span class="p4">(<span class="default"><a href="substut1.html#K13" title="SUBSTUT1:func.13">upVar</a> <span class="p5">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p0">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span>,<font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p1">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <span class="p2">(<span class="default"><a href="substut1.html#K13" title="SUBSTUT1:func.13">upVar</a> <span class="p3">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p4">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span>,<font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T87" title="FUNCT_4:th.87">FUNCT_4:87</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">F</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<span class="p3">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <span class="p4">(<span class="default"><a href="substut1.html#K13" title="SUBSTUT1:func.13">upVar</a> <span class="p5">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p0">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span>,<font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/><a NAME="E4:10_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T23" title="RELAT_1:th.23">RELAT_1:23</a></i>;<br/><a NAME="E5:10_1"/><b>then </b>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="ref" href="substut2.html#T6" target="_self" title="SUBSTUT2:th.6">Th6</a></i>;<br/><a NAME="E6:10_1"/><b>then </b>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="ref" href="substut1.html#D2" title="SUBSTUT1:def.2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E7:10_1"/><b>then </b><i><font color="Green" title="E13">A4</font></i>: 
<span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T32" title="FUNCT_4:th.32">FUNCT_4:32</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">Sub1</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a>;<br/><a NAME="E9:10_1"/>
 <a href="substut1.html#K14" title="SUBSTUT1:func.14">ExpandSub</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="txt" href="#E1:10_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="substut1.html#D13" title="SUBSTUT1:def.13">SUBSTUT1:def 13</a></i>;<br/><a NAME="E10:10_1"/><b>then </b>
 <a href="substut1.html#K14" title="SUBSTUT1:func.14">ExpandSub</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="txt" href="#E7:10_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="substut1.html#D2" title="SUBSTUT1:def.2">SUBSTUT1:def 2</a></i>;<br/><b>hence </b><a NAME="E11:10_1"/>
 <a href="substut1.html#K14" title="SUBSTUT1:func.14">ExpandSub</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3"><a href="substut2.html#K1" title="SUBSTUT2:func.1">[</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span><a href="substut2.html#K1" title="SUBSTUT2:func.1">]</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:10_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="#E2:10_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="substut2.html#T7" target="_self" title="SUBSTUT2:th.7">Th7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_2"/><i><font color="Green" title="E11">A6</font></i>: 
not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="substut1.html#K14" title="SUBSTUT1:func.14">ExpandSub</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3"><a href="substut2.html#K1" title="SUBSTUT2:func.1">[</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span><a href="substut2.html#K1" title="SUBSTUT2:func.1">]</a></span></span>)</span></span>)</span></span><br/><a NAME="E2:10_2"/><i><font color="Green" title="E12">A7</font></i>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T87" title="FUNCT_4:th.87">FUNCT_4:87</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">F</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/><a NAME="E4:10_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T23" title="RELAT_1:th.23">RELAT_1:23</a></i>;<br/><a NAME="E5:10_2"/><b>then </b>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="ref" href="substut2.html#T6" target="_self" title="SUBSTUT2:th.6">Th6</a></i>;<br/><a NAME="E6:10_2"/><b>then </b>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="ref" href="substut1.html#D2" title="SUBSTUT1:def.2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E7:10_2"/><b>then </b><i><font color="Green" title="E13">A8</font></i>: 
<span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T32" title="FUNCT_4:th.32">FUNCT_4:32</a></i>;<br/><a NAME="E8:10_2"/>
 <a href="substut1.html#K14" title="SUBSTUT1:func.14">ExpandSub</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="txt" href="#E1:10_2"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="substut1.html#D13" title="SUBSTUT1:def.13">SUBSTUT1:def 13</a></i>;<br/><a NAME="E9:10_2"/><b>then </b>
 <a href="substut1.html#K14" title="SUBSTUT1:func.14">ExpandSub</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="txt" href="#E7:10_2"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="substut1.html#D2" title="SUBSTUT1:def.2">SUBSTUT1:def 2</a></i>;<br/><b>hence </b><a NAME="E10:10_2"/>
 <a href="substut1.html#K14" title="SUBSTUT1:func.14">ExpandSub</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3"><a href="substut2.html#K1" title="SUBSTUT2:func.1">[</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span><a href="substut2.html#K1" title="SUBSTUT2:func.1">]</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:10_2"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="#E2:10_2"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="substut2.html#T8" target="_self" title="SUBSTUT2:th.8">Th8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:10"/>
 <a href="substut1.html#K14" title="SUBSTUT1:func.14">ExpandSub</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3"><a href="substut2.html#K1" title="SUBSTUT2:func.1">[</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">Sub</font></span><a href="substut2.html#K1" title="SUBSTUT2:func.1">]</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E10">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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