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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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b2">J</font> being    <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Olive" title="b1">A</font><br/>  for <font color="Olive" title="b3">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="b4">xSQ</font> being    <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Olive" title="b3">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>  st <span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Olive" title="b3">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>  &amp; (  for <font color="Olive" title="b5">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Olive" title="b1">A</font> holds <br/> ( <font color="Olive" title="b2">J</font>,<font color="Olive" title="b5">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <font color="Olive" title="b3">S</font> iff <font color="Olive" title="b2">J</font>,<font color="Olive" title="b5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> (<font color="Olive" title="b5">v</font>,<font color="Olive" title="b3">S</font>)</span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Olive" title="b3">S</font> ) ) holds <br/> for <font color="Olive" title="b5">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Olive" title="b1">A</font> holds <br/> ( <font color="Olive" title="b2">J</font>,<font color="Olive" title="b5">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p1">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p2"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Olive" title="b3">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Olive" title="b4">xSQ</font>)</span>)</span> iff <font color="Olive" title="b2">J</font>,<font color="Olive" title="b5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> (<font color="Olive" title="b5">v</font>,<span class="p2">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p3"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Olive" title="b3">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Olive" title="b4">xSQ</font>)</span>)</span>)</span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a>  <a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Olive" title="b3">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Olive" title="b4">xSQ</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">J</font> being    <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c2">A</font><br/>  for <font color="Olive" title="b2">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="b3">xSQ</font> being    <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Olive" title="b2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>  st <span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Olive" title="b2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>  &amp; (  for <font color="Olive" title="b4">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c2">A</font> holds <br/> ( <font color="Olive" title="b1">J</font>,<font color="Olive" title="b4">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <font color="Olive" title="b2">S</font> iff <font color="Olive" title="b1">J</font>,<font color="Olive" title="b4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> (<font color="Olive" title="b4">v</font>,<font color="Olive" title="b2">S</font>)</span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Olive" title="b2">S</font> ) ) holds <br/> for <font color="Olive" title="b4">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c2">A</font> holds <br/> ( <font color="Olive" title="b1">J</font>,<font color="Olive" title="b4">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p1">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p2"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Olive" title="b2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Olive" title="b3">xSQ</font>)</span>)</span> iff <font color="Olive" title="b1">J</font>,<font color="Olive" title="b4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> (<font color="Olive" title="b4">v</font>,<span class="p2">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p3"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Olive" title="b2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Olive" title="b3">xSQ</font>)</span>)</span>)</span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a>  <a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Olive" title="b2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Olive" title="b3">xSQ</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">J</font> be    <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>  st <span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>  &amp; (  for <font color="Olive" title="b3">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c2">A</font> holds <br/> ( <font color="Maroon" title="c3">J</font>,<font color="Olive" title="b3">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <font color="Olive" title="b1">S</font> iff <font color="Maroon" title="c3">J</font>,<font color="Olive" title="b3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> (<font color="Olive" title="b3">v</font>,<font color="Olive" title="b1">S</font>)</span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Olive" title="b1">S</font> ) ) holds <br/> for <font color="Olive" title="b3">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c2">A</font> holds <br/> ( <font color="Maroon" title="c3">J</font>,<font color="Olive" title="b3">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p1">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p2"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Olive" title="b2">xSQ</font>)</span>)</span> iff <font color="Maroon" title="c3">J</font>,<font color="Olive" title="b3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> (<font color="Olive" title="b3">v</font>,<span class="p2">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p3"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Olive" title="b2">xSQ</font>)</span>)</span>)</span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a>  <a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Olive" title="b2">xSQ</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">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="firebrick">::  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#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>  st <span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>  &amp; (  for <font color="Olive" title="b2">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c2">A</font> holds <br/> ( <font color="Maroon" title="c3">J</font>,<font color="Olive" title="b2">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <font color="Maroon" title="c4">S</font> iff <font color="Maroon" title="c3">J</font>,<font color="Olive" title="b2">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> (<font color="Olive" title="b2">v</font>,<font color="Maroon" title="c4">S</font>)</span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c4">S</font> ) ) holds <br/> for <font color="Olive" title="b2">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c2">A</font> holds <br/> ( <font color="Maroon" title="c3">J</font>,<font color="Olive" title="b2">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p1">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p2"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Olive" title="b1">xSQ</font>)</span>)</span> iff <font color="Maroon" title="c3">J</font>,<font color="Olive" title="b2">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> (<font color="Olive" title="b2">v</font>,<span class="p2">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p3"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Olive" title="b1">xSQ</font>)</span>)</span>)</span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a>  <a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Olive" title="b1">xSQ</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">xSQ</font> be    <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>  &amp; (  for <font color="Olive" title="b1">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c2">A</font> holds <br/> ( <font color="Maroon" title="c3">J</font>,<font color="Olive" title="b1">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <font color="Maroon" title="c4">S</font> iff <font color="Maroon" title="c3">J</font>,<font color="Olive" title="b1">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> (<font color="Olive" title="b1">v</font>,<font color="Maroon" title="c4">S</font>)</span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c4">S</font> ) ) implies  for <font color="Olive" title="b1">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c2">A</font> holds <br/> ( <font color="Maroon" title="c3">J</font>,<font color="Olive" title="b1">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p1">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p2"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span> iff <font color="Maroon" title="c3">J</font>,<font color="Olive" title="b1">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> (<font color="Olive" title="b1">v</font>,<span class="p2">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p3"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span>)</span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a>  <a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>) ) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:109"/><span class="lab"><font color="Green" title="E88">A1</font></span>: 
<span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a> 
 <span class="kw">and </span><br/><a NAME="E2:109"/><span class="lab"><font color="Green" title="E89">A2</font></span>: 
 for <font color="Olive" title="b1">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c2">A</font> holds <br/> ( <font color="Maroon" title="c3">J</font>,<font color="Olive" title="b1">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <font color="Maroon" title="c4">S</font> iff <font color="Maroon" title="c3">J</font>,<font color="Olive" title="b1">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> (<font color="Olive" title="b1">v</font>,<font color="Maroon" title="c4">S</font>)</span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c4">S</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c2">A</font> holds <br/> ( <font color="Maroon" title="c3">J</font>,<font color="Olive" title="b1">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p1">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p2"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span> iff <font color="Maroon" title="c3">J</font>,<font color="Olive" title="b1">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> (<font color="Olive" title="b1">v</font>,<span class="p2">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p3"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span>)</span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a>  <a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">v</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p1">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p2"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span> iff <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> (<font color="Maroon" title="c6">v</font>,<span class="p2">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p3"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span>)</span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a>  <a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">S1</font> =  <a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c8">z</font> =  <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#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p3"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span>;<br/>
<a NAME="E3:109"/><span class="lab"><font color="Green" title="E90">A3</font></span>: 
( (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p0"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">S</font>)</span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c4">S</font> ) iff  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p0"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K13" title="SUBLEMMA:func.13">NEx_Val</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p0">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p1">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p2"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span> <a href="sublemma.html#K14" title="SUBLEMMA:func.14">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c4">S</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:109"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="ref" href="sublemma.html#T54" target="_self" title="SUBLEMMA:th.54">Th55</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">q</font> =  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <font color="Maroon" title="c4">S</font>;<br/>
<a NAME="E4:109"/><span class="lab"><font color="Green" title="E91">A4</font></span>: 
( <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<span class="p1">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p2">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p3">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p4"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <font color="Maroon" title="c4">S</font></span>)</span>) iff  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p4">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p5"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <font color="Maroon" title="c4">S</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sublemma.html#T50" target="_self" title="SUBLEMMA:th.50">Th51</a></span>;<br/>
<a NAME="E5:109"/><span class="lab"><font color="Green" title="E92">A5</font></span>: 
( (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p4">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p5"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <font color="Maroon" title="c4">S</font> ) implies  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p0"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">S</font>)</span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c4">S</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:109_1"/><span class="lab"><font color="Green" title="E93">A6</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p4">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p5"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <font color="Maroon" title="c4">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p0"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">S</font>)</span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c4">S</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c10">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p0"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Maroon" title="c10">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Maroon" title="c10">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">S</font>)</span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c4">S</font></span><br/>

<a NAME="E2:109_1"/>
<font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p4">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p5"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <font color="Maroon" title="c4">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:109_1"><span class="lab"><font color="Green" title="E93">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:109_1"/>
<font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p0"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Maroon" title="c10">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Maroon" title="c10">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">S</font>)</span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c4">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109"><span class="lab"><font color="Green" title="E89">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:109"/><span class="lab"><font color="Green" title="E93">A7</font></span>: 
( (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p0"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">S</font>)</span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c4">S</font> ) implies  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p4">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p5"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <font color="Maroon" title="c4">S</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:109_2"/><span class="lab"><font color="Green" title="E94">A8</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p0"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">S</font>)</span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c4">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p4">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p5"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <font color="Maroon" title="c4">S</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c10">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p4">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p5"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <font color="Maroon" title="c4">S</font></span><br/>

<a NAME="E2:109_2"/>
<font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p0"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Maroon" title="c10">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Maroon" title="c10">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">S</font>)</span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c4">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:109_2"><span class="lab"><font color="Green" title="E94">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:109_2"/>
<font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p4">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p5"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <font color="Maroon" title="c4">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109"><span class="lab"><font color="Green" title="E89">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c10">p</font> =  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p1">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_the_scope_of</a> <span class="p2">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p3"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span>;<br/>
<a NAME="E7:109"/><span class="lab"><font color="Green" title="E94">A9</font></span>: 
( <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="sublemma.html#K11" title="SUBLEMMA:func.11">CQCQuant</a> (<span class="p1">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p2"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p2">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_the_scope_of</a> <span class="p3">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p4"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span>) iff <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="sublemma.html#K11" title="SUBLEMMA:func.11">CQCQuant</a> (<span class="p1">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p2"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <font color="Maroon" title="c4">S</font></span>)</span>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:109"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="ref" href="sublemma.html#T30" target="_self" title="SUBLEMMA:th.30">Th31</a></span>;<br/>
<a NAME="E8:109"/><span class="lab"><font color="Green" title="E95">A10</font></span>: 
( (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p0"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K13" title="SUBLEMMA:func.13">NEx_Val</a> (<font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span> <a href="sublemma.html#K14" title="SUBLEMMA:func.14">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c4">S</font> ) iff  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K13" title="SUBLEMMA:func.13">NEx_Val</a> (<font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span> <a href="sublemma.html#K14" title="SUBLEMMA:func.14">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c4">S</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:109"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="ref" href="sublemma.html#T70" target="_self" title="SUBLEMMA:th.70">Th72</a></span>;<br/>
<a NAME="E9:109"/><span class="lab"><font color="Green" title="E96">A11</font></span>: 
( <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K13" title="SUBLEMMA:func.13">NEx_Val</a> (<font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a></span>)</span>) implies  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K13" title="SUBLEMMA:func.13">NEx_Val</a> (<font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c4">S</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:109_3"/>
<font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K13" title="SUBLEMMA:func.13">NEx_Val</a> (<font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a></span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K13" title="SUBLEMMA:func.13">NEx_Val</a> (<font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c4">S</font></span><br/>

<a NAME="E2:109_3"/><span class="kw">then </span>
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K13" title="SUBLEMMA:func.13">NEx_Val</a> (<font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sublemma.html#T50" target="_self" title="SUBLEMMA:th.50">Th51</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:109_3"/>
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K13" title="SUBLEMMA:func.13">NEx_Val</a> (<font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c4">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sublemma.html#T73" target="_self" title="SUBLEMMA:th.73">Th76</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:109"/><span class="lab"><font color="Green" title="E97">A12</font></span>: 
( (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K13" title="SUBLEMMA:func.13">NEx_Val</a> (<font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c4">S</font> ) implies <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K13" title="SUBLEMMA:func.13">NEx_Val</a> (<font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a></span>)</span>) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:109_4"/>
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K13" title="SUBLEMMA:func.13">NEx_Val</a> (<font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c4">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K13" title="SUBLEMMA:func.13">NEx_Val</a> (<font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a></span>)</span>)</span><br/>

<a NAME="E2:109_4"/><span class="kw">then </span>
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c3">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K13" title="SUBLEMMA:func.13">NEx_Val</a> (<font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sublemma.html#T73" target="_self" title="SUBLEMMA:th.73">Th76</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:109_4"/>
<font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K13" title="SUBLEMMA:func.13">NEx_Val</a> (<font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">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="ref" href="sublemma.html#T50" target="_self" title="SUBLEMMA:th.50">Th51</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:109"/>
 <a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>) is  <a href="substut1.html#V7" title="SUBSTUT1:attr.7">Sub_universal</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:109"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="ref" href="sublemma.html#T27" target="_self" title="SUBLEMMA:th.27">Th28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:109"/>
( <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p1">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p2"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span> iff <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> (<font color="Maroon" title="c6">v</font>,<span class="p2">(<span class="default"><a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p3"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>)</span>)</span>)</span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a>  <a href="sublemma.html#K9" title="SUBLEMMA:func.9">CQCSub_All</a> (<span class="p1"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K7" title="SUBLEMMA:func.7">]</a></span>,<font color="Maroon" title="c5">xSQ</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:109"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E7:109"><span class="lab"><font color="Green" title="E94">A9</font></span></a>, <a class="txt" href="#E4:109"><span class="lab"><font color="Green" title="E91">A4</font></span></a>, <a class="txt" href="#E5:109"><span class="lab"><font color="Green" title="E92">A5</font></span></a>, <a class="txt" href="#E6:109"><span class="lab"><font color="Green" title="E93">A7</font></span></a>, <a class="txt" href="#E3:109"><span class="lab"><font color="Green" title="E90">A3</font></span></a>, <a class="txt" href="#E8:109"><span class="lab"><font color="Green" title="E95">A10</font></span></a>, <a class="txt" href="#E10:109"><span class="lab"><font color="Green" title="E97">A12</font></span></a>, <a class="txt" href="#E9:109"><span class="lab"><font color="Green" title="E96">A11</font></span></a>, <a class="ref" href="sublemma.html#T28" target="_self" title="SUBLEMMA:th.28">Th29</a>, <a class="ref" href="sublemma.html#T31" target="_self" title="SUBLEMMA:th.31">Th32</a>, <a class="ref" href="sublemma.html#T56" target="_self" title="SUBLEMMA:th.56">Th57</a>, <a class="ref" href="sublemma.html#T72" target="_self" title="SUBLEMMA:th.72">Th75</a>, <a class="ref" href="sublemma.html#T87" target="_self" title="SUBLEMMA:th.87">Th91</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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