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<span class="kw">let </span><font color="Maroon" title="c1">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <br/> for <font color="Olive" title="b2">xSQ</font> being   <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>  st <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>  &amp; <font color="Maroon" title="c1">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="c1">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="c1">x</font>,<span class="p3">(<span class="default"><font color="Olive" title="b1">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Olive" title="b2">xSQ</font></span>)</span> holds <br/>( not  <a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p1">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p2">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Olive" title="b2">xSQ</font></span>)</span></span>)</span> <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="c1">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="c1">x</font>,<span class="p3">(<span class="default"><font color="Olive" title="b1">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Olive" title="b2">xSQ</font></span>)</span> &amp; not  <a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p1">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p2">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Olive" title="b2">xSQ</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">xSQ</font> being   <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>  st <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>  &amp; <font color="Maroon" title="c1">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="c1">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="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Olive" title="b1">xSQ</font></span>)</span> holds <br/>( not  <a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p1">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p2">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Olive" title="b1">xSQ</font></span>)</span></span>)</span> <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="c1">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="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Olive" title="b1">xSQ</font></span>)</span> &amp; not  <a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p1">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p2">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Olive" title="b1">xSQ</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">xSQ</font> be    <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>  &amp; <font color="Maroon" title="c1">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="c1">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="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> implies ( not  <a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p1">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p2">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span></span>)</span> <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="c1">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="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> &amp; not  <a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p1">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p2">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span> ) )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">S1</font> =  <a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:55"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
<span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a> 
 <span class="kw">and </span><br/><a NAME="E2:55"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon" title="c1">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="c1">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="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c3">xSQ</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not  <a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p1">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p2">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span></span>)</span> <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="c1">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="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> &amp; not  <a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p1">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p2">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span> )</span><br/>

<a NAME="E3:55"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
 <a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="sublemma.html#D6" target="_self" title="SUBLEMMA:def.6">Def6</a></span>;<br/>
<a NAME="E4:55"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <span class="p1">(<span class="default"><span class="p2"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span> <a href="substut1.html#K23" title="SUBSTUT1:func.23">`2</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span> <a href="sublemma.html#K9" title="SUBLEMMA:func.9">`1</a> </span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="sublemma.html#T27" target="_self" title="SUBLEMMA:th.27">Th27</a></span>;<br/>
<a NAME="E5:55"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<span class="p1">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span> <a href="sublemma.html#K9" title="SUBLEMMA:func.9">`1</a> </span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E6:55"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A5</font></span>: 
 <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#T8" title="QC_LANG2:th.8">QC_LANG2:8</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">finSub</font> =  <a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span>;<br/>
<a NAME="E7:55"/><span class="lab"><font color="Green" title="E44">A6</font></span>: 
 <a href="substut1.html#K10" title="SUBSTUT1:func.10">Dom_Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</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>  :  <a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="substut1.html#D9" title="SUBSTUT1:def.9">SUBSTUT1:def 9</a></span>;<br/>
<a NAME="E8:55"/>
<span class="p1">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</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="c3">xSQ</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E3:55"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="sublemma.html#T27" target="_self" title="SUBLEMMA:th.27">Th27</a></span>;<br/>
<a NAME="E9:55"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A7</font></span>: 
 <a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">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="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c3">xSQ</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:55"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E6:55"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">Y</font> = <span class="p1">(<span class="default"><a href="substut1.html#K10" title="SUBSTUT1:func.10">Dom_Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="substut1.html#K11" title="SUBSTUT1:func.11">Sub_Var</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">n</font> =  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <span class="p1">(<span class="default"><a href="substut1.html#K12" title="SUBSTUT1:func.12">NSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E10:55"/><span class="lab"><font color="Green" title="E46">A8</font></span>: 
 <a href="substut1.html#K13" title="SUBSTUT1:func.13">upVar</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <span class="p1">(<span class="default"><a href="substut1.html#K12" title="SUBSTUT1:func.12">NSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="substut1.html#D12" title="SUBSTUT1:def.12">SUBSTUT1:def 12</a></span>;<br/>
<a NAME="E11:55"/>
 <a href="substut1.html#K12" title="SUBSTUT1:func.12">NSub</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K10" title="SUBSTUT1:func.10">Dom_Bound_Vars</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="substut1.html#K11" title="SUBSTUT1:func.11">Sub_Var</a> <span class="p3">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <span class="p4">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="substut1.html#D11" title="SUBSTUT1:def.11">SUBSTUT1:def 11</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">X</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K10" title="SUBSTUT1:func.10">Dom_Bound_Vars</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="substut1.html#K11" title="SUBSTUT1:func.11">Sub_Var</a> <span class="p3">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <span class="p4">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E13:55"/><span class="lab"><font color="Green" title="E47">A9</font></span>: 
 <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <span class="p1">(<span class="default"><a href="substut1.html#K12" title="SUBSTUT1:func.12">NSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c8">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="substut1.html#D11" title="SUBSTUT1:def.11">SUBSTUT1:def 11</a></span>;<br/>
<a NAME="E14:55"/>
(  <a href="substut1.html#K10" title="SUBSTUT1:func.10">Dom_Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="substut1.html#K10" title="SUBSTUT1:func.10">Dom_Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="substut1.html#K11" title="SUBSTUT1:func.11">Sub_Var</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span></span>)</span> &amp;  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c8">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K10" title="SUBSTUT1:func.10">Dom_Bound_Vars</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="substut1.html#K11" title="SUBSTUT1:func.11">Sub_Var</a> <span class="p3">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <span class="p4">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xxreal_2.html#D7" title="XXREAL_2:def.7">XXREAL_2:def 7</a></span>;<br/>
<a NAME="E15:55"/><span class="kw">then </span>
not  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <span class="p1">(<span class="default"><a href="substut1.html#K12" title="SUBSTUT1:func.12">NSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K10" title="SUBSTUT1:func.10">Dom_Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:55"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E16:55"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A10</font></span>: 
not  <a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <span class="p2">(<span class="default"><a href="substut1.html#K12" title="SUBSTUT1:func.12">NSub</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <span class="p4">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:55"><span class="lab"><font color="Green" title="E44">A6</font></span></a></span>;<br/>
<a NAME="E17:55"/>
<span class="p1">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:55"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E18:55"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A11</font></span>: 
not  <a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <span class="p1">(<span class="default"><a href="substut1.html#K13" title="SUBSTUT1:func.13">upVar</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:55"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="txt" href="#E16:55"><span class="lab"><font color="Green" title="E48">A10</font></span></a>, <a class="ref" href="qc_lang2.html#T8" title="QC_LANG2:th.8">QC_LANG2:8</a></span>;<br/>
<a NAME="E19:55"/>
(  <a href="substut1.html#K11" title="SUBSTUT1:func.11">Sub_Var</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="substut1.html#K10" title="SUBSTUT1:func.10">Dom_Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="substut1.html#K11" title="SUBSTUT1:func.11">Sub_Var</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span></span>)</span> &amp;  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c8">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K10" title="SUBSTUT1:func.10">Dom_Bound_Vars</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="substut1.html#K11" title="SUBSTUT1:func.11">Sub_Var</a> <span class="p3">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <span class="p4">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xxreal_2.html#D7" title="XXREAL_2:def.7">XXREAL_2:def 7</a></span>;<br/>
<a NAME="E20:55"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A12</font></span>: 
not  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <span class="p1">(<span class="default"><a href="substut1.html#K12" title="SUBSTUT1:func.12">NSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K11" title="SUBSTUT1:func.11">Sub_Var</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:55"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E21:55"/>
 <a href="substut1.html#K11" title="SUBSTUT1:func.11">Sub_Var</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</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>  :  <a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <font color="Olive" title="b1">i</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> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="substut1.html#D10" title="SUBSTUT1:def.10">SUBSTUT1:def 10</a></span>;<br/>
<a NAME="E22:55"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A13</font></span>: 
not  <a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <span class="p1">(<span class="default"><a href="substut1.html#K13" title="SUBSTUT1:func.13">upVar</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:55"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="txt" href="#E20:55"><span class="lab"><font color="Green" title="E50">A12</font></span></a></span>;<br/>
<a NAME="E23:55"/>
 <a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p1">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="substut1.html#D35" title="SUBSTUT1:def.35">SUBSTUT1:def 35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:55"/>
( not  <a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p1">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p2">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span></span>)</span> <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="c1">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="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c3">xSQ</font></span>)</span> &amp; not  <a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p1">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p2">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E6:55"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="txt" href="#E9:55"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="txt" href="#E22:55"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E18:55"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="ref" href="substut1.html#D36" title="SUBSTUT1:def.36">SUBSTUT1:def 36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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