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<span class="kw">let </span><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font> being  <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c1">k</font><br/> for <font color="Olive" title="b2">l</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</font><br/> for <font color="Olive" title="b3">Sub</font> being  <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> holds   <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Olive" title="b2">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p2"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Olive" title="b2">l</font></span>)</span>,<font color="Olive" title="b3">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">P</font> be   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c1">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">l</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</font><br/> for <font color="Olive" title="b2">Sub</font> being  <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> holds   <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p2"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Olive" title="b1">l</font></span>)</span>,<font color="Olive" title="b2">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">l</font> be   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Sub</font> being  <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> holds   <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p2"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">l</font></span>)</span>,<font color="Olive" title="b1">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">Sub</font> be   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p2"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">l</font></span>)</span>,<font color="Maroon" title="c4">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span></span>)</span></span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">S</font> = <span class="p1"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">l</font></span>)</span>,<font color="Maroon" title="c4">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span>;<br/>
<a NAME="E1:22"/>
<span class="p1"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">l</font></span>)</span>,<font color="Maroon" title="c4">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">Sub</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="substut1.html#T9" title="SUBSTUT1:th.9">SUBSTUT1:9</a></span>;<br/>
<a NAME="E2:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A1</font></span>: 
 <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p1"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">l</font></span>)</span>,<font color="Maroon" title="c4">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <span class="p1">(<span class="default"><a href="sublemma.html#K5" title="SUBLEMMA:func.5">CQC_Subst</a> <font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">Sub</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sublemma.html#T9" title="SUBLEMMA:th.9">SUBLEMMA:9</a></span>;<br/>
<a NAME="E3:22"/>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <span class="p2">(<span class="default"><a href="sublemma.html#K5" title="SUBLEMMA:func.5">CQC_Subst</a> <font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">Sub</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_sim1.html#T15" title="CQC_SIM1:th.15">CQC_SIM1:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:22"/>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p2"><a href="substut2.html#K2" title="SUBSTUT2:func.2">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">l</font></span>)</span>,<font color="Maroon" title="c4">Sub</font></span><a href="substut2.html#K2" title="SUBSTUT2:func.2">]</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="cqc_sim1.html#T15" title="CQC_SIM1:th.15">CQC_SIM1:15</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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