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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="firebrick">::  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">ll</font> being   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</font><br/>  for <font color="Olive" title="b3">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> holds   <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p1">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> (<font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">ll</font>,<font color="Olive" title="b3">Sub</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <span class="p1">(<span class="default"><a href="substut1.html#K3" title="SUBSTUT1:func.3">CQC_Subst</a> (<font color="Olive" title="b2">ll</font>,<font color="Olive" title="b3">Sub</font>)</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">ll</font> being   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</font><br/>  for <font color="Olive" title="b2">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> holds   <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p1">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> (<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b1">ll</font>,<font color="Olive" title="b2">Sub</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <span class="p1">(<span class="default"><a href="substut1.html#K3" title="SUBSTUT1:func.3">CQC_Subst</a> (<font color="Olive" title="b1">ll</font>,<font color="Olive" title="b2">Sub</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">ll</font> be   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p1">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> (<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Olive" title="b1">Sub</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <span class="p1">(<span class="default"><a href="substut1.html#K3" title="SUBSTUT1:func.3">CQC_Subst</a> (<font color="Maroon" title="c3">ll</font>,<font color="Olive" title="b1">Sub</font>)</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="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p1">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> (<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">Sub</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <span class="p1">(<span class="default"><a href="substut1.html#K3" title="SUBSTUT1:func.3">CQC_Subst</a> (<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">Sub</font>)</span>)</span></span><br/>







<a NAME="E1:15"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c3">ll</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#D16" title="QC_LANG1:def.16">QC_LANG1:def 16</a></span>;<br/>
<a NAME="E2:15"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
 <a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> (<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">Sub</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c3">ll</font></span>)</span>,<font color="Maroon" title="c4">Sub</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<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="E3:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<span class="p1">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> (<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">Sub</font>)</span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Sub</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E4:15"/>
<span class="p1">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> (<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">Sub</font>)</span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c3">ll</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E5:15"/><span class="kw">then </span>
(  <a href="substut1.html#K26" title="SUBSTUT1:func.26">Sub_the_arguments_of</a> <span class="p1">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> (<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">Sub</font>)</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">ll</font> &amp;  <a href="qc_lang1.html#K15" title="QC_LANG1:func.15">the_pred_symbol_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> (<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">Sub</font>)</span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="qc_lang1.html#D20" title="QC_LANG1:def.20">QC_LANG1:def 20</a>, <a class="ref" href="substut1.html#D29" title="SUBSTUT1:def.29">SUBSTUT1:def 29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:15"/>
 <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <span class="p1">(<span class="default"><a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> (<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">Sub</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <span class="p1">(<span class="default"><a href="substut1.html#K3" title="SUBSTUT1:func.3">CQC_Subst</a> (<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">Sub</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="sublemma.html#T6" target="_self" title="SUBLEMMA:th.6">Th6</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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