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<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> ; <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">A</font> being 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/> for <font color="Olive" title="b2">v</font> being   <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="Olive" title="b1">A</font><br/> for <font color="Olive" title="b3">P</font> being  <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c1">k</font><br/> for <font color="Olive" title="b4">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</font><br/> for <font color="Olive" title="b5">Sub</font> being  <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> holds  <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Olive" title="b2">v</font>,<span class="p3">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Olive" title="b3">P</font>,<font color="Olive" title="b4">ll</font>,<font color="Olive" title="b5">Sub</font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Olive" title="b4">ll</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <span class="p1">(<span class="default"><a href="sublemma.html#K5" title="SUBLEMMA:func.5">CQC_Subst</a> <font color="Olive" title="b4">ll</font>,<font color="Olive" title="b5">Sub</font></span>)</span></span><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> ; <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">v</font> being   <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/> for <font color="Olive" title="b2">P</font> being  <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c1">k</font><br/> for <font color="Olive" title="b3">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</font><br/> for <font color="Olive" title="b4">Sub</font> being  <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Olive" title="b1">v</font>,<span class="p3">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Olive" title="b2">P</font>,<font color="Olive" title="b3">ll</font>,<font color="Olive" title="b4">Sub</font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Olive" title="b3">ll</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <span class="p1">(<span class="default"><a href="sublemma.html#K5" title="SUBLEMMA:func.5">CQC_Subst</a> <font color="Olive" title="b3">ll</font>,<font color="Olive" title="b4">Sub</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">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>; <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">P</font> being  <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c1">k</font><br/> for <font color="Olive" title="b2">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</font><br/> for <font color="Olive" title="b3">Sub</font> being  <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p3">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">ll</font>,<font color="Olive" title="b3">Sub</font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Olive" title="b2">ll</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <span class="p1">(<span class="default"><a href="sublemma.html#K5" title="SUBLEMMA:func.5">CQC_Subst</a> <font color="Olive" title="b2">ll</font>,<font color="Olive" title="b3">Sub</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">P</font> be   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c1">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</font><br/> for <font color="Olive" title="b2">Sub</font> being  <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p3">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Olive" title="b1">ll</font>,<font color="Olive" title="b2">Sub</font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Olive" title="b1">ll</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <span class="p1">(<span class="default"><a href="sublemma.html#K5" title="SUBLEMMA:func.5">CQC_Subst</a> <font color="Olive" title="b1">ll</font>,<font color="Olive" title="b2">Sub</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">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>; <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  <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p3">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Olive" title="b1">Sub</font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c5">ll</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <span class="p1">(<span class="default"><a href="sublemma.html#K5" title="SUBLEMMA:func.5">CQC_Subst</a> <font color="Maroon" title="c5">ll</font>,<font color="Olive" title="b1">Sub</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c6">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"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p3">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c5">ll</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <span class="p1">(<span class="default"><a href="sublemma.html#K5" title="SUBLEMMA:func.5">CQC_Subst</a> <font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span><br/>











<b>set </b><font color="Maroon" title="c7">S'</font> =  <a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font>;<br/>
<a NAME="E1:21"/>
 <a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c5">ll</font></span>)</span>,<font color="Maroon" title="c6">Sub</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="ref" href="substut1.html#T9" title="SUBSTUT1:th.9">SUBSTUT1:9</a></i>;<br/>
<a NAME="E2:21"/><b>then </b><i><font color="Green" title="E13">A1</font></i>: 
<span class="p1">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Sub</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">ll'</font> =  <a href="sublemma.html#K5" title="SUBLEMMA:func.5">CQC_Subst</a> <font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font>;<br/>
<a NAME="E3:21"/><i><font color="Green" title="E14">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">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="qc_lang1.html#D8" title="QC_LANG1:def.8">QC_LANG1:def 8</a></i>;<br/>
<a NAME="E4:21"/><i><font color="Green" title="E15">Y</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <span class="p2">(<span class="default"><a href="sublemma.html#K5" title="SUBLEMMA:func.5">CQC_Subst</a> <font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></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="E5:21"/><i><font color="Green" title="E16">A14</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="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p4">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c5">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="E6:21"/><b>then </b><i><font color="Green" title="E17">X</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="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p4">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c5">ll</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:21"/>
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</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="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p4">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c5">ll</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p4">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c5">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <span class="p2">(<span class="default"><a href="sublemma.html#K5" title="SUBLEMMA:func.5">CQC_Subst</a> <font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">j</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="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p4">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c5">ll</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p4">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c5">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <span class="p2">(<span class="default"><a href="sublemma.html#K5" title="SUBLEMMA:func.5">CQC_Subst</a> <font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> )</span><br/>

<b>assume </b><a NAME="E1:21_1"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c9">j</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="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p4">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c5">ll</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p4">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c5">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <span class="p2">(<span class="default"><a href="sublemma.html#K5" title="SUBLEMMA:func.5">CQC_Subst</a> <font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span><br/>

<a NAME="E2:21_1"/><i><font color="Green" title="E19">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font> &amp; <font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> )
 
<b>by </b><i><a class="txt" href="#E1:21_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E6:21"><i><font color="Green" title="E17">X</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">j</font> = <font color="Maroon" title="c9">j</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E4:21_1"/>
<font color="Maroon" title="c10">j</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">ll</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:21_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E6:21"><i><font color="Green" title="E17">X</font></i></a>, <a class="ref" href="qc_lang1.html#D8" title="QC_LANG1:def.8">QC_LANG1:def 8</a></i>;<br/>
<a NAME="E5:21_1"/><b>then </b>
<font color="Maroon" title="c10">j</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">ll</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">x</font> = <font color="Maroon" title="c5">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> as   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> <b>by </b><i><a class="ref" href="sublemma.html#T5" target="_self" title="SUBLEMMA:th.5">Th5</a></i>;<br/>
<div><i><font color="Green" title="E20">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_1"/><i><font color="Green" title="E21">A7</font></i>: 
not <font color="Maroon" title="c5">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Sub</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p4">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c5">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <span class="p2">(<span class="default"><a href="sublemma.html#K5" title="SUBLEMMA:func.5">CQC_Subst</a> <font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span><br/><a NAME="E2:21_1_1"/><b>then </b><i><font color="Green" title="E22">A8</font></i>: 
<font color="Maroon" title="c3">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K5" title="SUBLEMMA:func.5">CQC_Subst</a> <font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E2:21_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="substut1.html#D3" title="SUBSTUT1:def.3">SUBSTUT1:def 3</a></i>;<br/><a NAME="E3:21_1_1"/><i><font color="Green" title="E23">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <span class="p2">(<span class="default"><a href="sublemma.html#K5" title="SUBLEMMA:func.5">CQC_Subst</a> <font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K5" title="SUBLEMMA:func.5">CQC_Subst</a> <font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:21_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="valuat_1.html#D8" title="VALUAT_1:def.8">VALUAT_1:def 8</a></i>;<br/><a NAME="E4:21_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p3">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></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="c5">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="#E2:21"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="#E1:21_1_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="sublemma.html#T12" target="_self" title="SUBLEMMA:th.12">Th12</a></i>;<br/><b>hence </b><a NAME="E5:21_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p4">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c5">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <span class="p2">(<span class="default"><a href="sublemma.html#K5" title="SUBLEMMA:func.5">CQC_Subst</a> <font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="#E2:21_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="#E2:21_1_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="#E3:21_1_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="valuat_1.html#D8" title="VALUAT_1:def.8">VALUAT_1:def 8</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="21_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_2"/><i><font color="Green" title="E21">A10</font></i>: 
<font color="Maroon" title="c5">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Sub</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p4">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c5">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <span class="p2">(<span class="default"><a href="sublemma.html#K5" title="SUBLEMMA:func.5">CQC_Subst</a> <font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span><br/><a NAME="E2:21_1_2"/><b>then </b><i><font color="Green" title="E22">A11</font></i>: 
<font color="Maroon" title="c3">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K5" title="SUBLEMMA:func.5">CQC_Subst</a> <font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:21"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="#E3:21"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E2:21_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="substut1.html#D3" title="SUBSTUT1:def.3">SUBSTUT1:def 3</a></i>;<br/><a NAME="E3:21_1_2"/><i><font color="Green" title="E23">A12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p3">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></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="c5">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p2">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></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="#E2:21"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="#E1:21_1_2"><i><font color="Green" title="E21">A10</font></i></a>, <a class="ref" href="sublemma.html#T13" target="_self" title="SUBLEMMA:th.13">Th13</a></i>;<br/><a NAME="E4:21_1_2"/>
<font color="Maroon" title="c5">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</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"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:21"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="#E1:21_1_2"><i><font color="Green" title="E21">A10</font></i></a>, <a class="ref" href="substut1.html#D2" title="SUBSTUT1:def.2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E5:21_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p3">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></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="c5">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:21_1_2"><i><font color="Green" title="E23">A12</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E6:21_1_2"/><b>then </b><i><font color="Green" title="E24">A13</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p3">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></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="c5">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="substut1.html#D2" title="SUBSTUT1:def.2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E7:21_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p4">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c5">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p3">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></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="c5">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:21_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="valuat_1.html#D8" title="VALUAT_1:def.8">VALUAT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E8:21_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p4">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c5">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <span class="p2">(<span class="default"><a href="sublemma.html#K5" title="SUBLEMMA:func.5">CQC_Subst</a> <font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="#E2:21_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="#E2:21_1_2"><i><font color="Green" title="E22">A11</font></i></a>, <a class="txt" href="#E6:21_1_2"><i><font color="Green" title="E24">A13</font></i></a>, <a class="ref" href="valuat_1.html#D8" title="VALUAT_1:def.8">VALUAT_1:def 8</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="E9:21_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p4">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c5">ll</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <span class="p2">(<span class="default"><a href="sublemma.html#K5" title="SUBLEMMA:func.5">CQC_Subst</a> <font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="#E7:21_1"><i><font color="Green" title="E20">A6</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/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:21"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p3">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c5">ll</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <span class="p1">(<span class="default"><a href="sublemma.html#K5" title="SUBLEMMA:func.5">CQC_Subst</a> <font color="Maroon" title="c5">ll</font>,<font color="Maroon" title="c6">Sub</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:21"><i><font color="Green" title="E16">A14</font></i></a>, <a class="txt" href="#E4:21"><i><font color="Green" title="E15">Y</font></i></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</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>
