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<b>let </b><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">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">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a>  holds   <a href="substut1.html#K37" title="SUBSTUT1:func.37">CQC_Sub</a> <span class="p1">(<span class="default"><a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">ll</font>,<font color="Olive" title="b3">e</font></span>)</span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> </span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</font><br/> for <font color="Olive" title="b2">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a>  holds   <a href="substut1.html#K37" title="SUBSTUT1:func.37">CQC_Sub</a> <span class="p1">(<span class="default"><a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Olive" title="b1">ll</font>,<font color="Olive" title="b2">e</font></span>)</span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> </span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a>  holds   <a href="substut1.html#K37" title="SUBSTUT1:func.37">CQC_Sub</a> <span class="p1">(<span class="default"><a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Olive" title="b1">e</font></span>)</span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> </span><br/><b>let </b><font color="Maroon" title="c4">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="substut1.html#K37" title="SUBSTUT1:func.37">CQC_Sub</a> <span class="p1">(<span class="default"><a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">e</font></span>)</span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> </span><br/>







<b>consider </b><font color="Maroon" title="c5">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> , <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> <b> such that </b><br/><a NAME="E2:92"/><i><font color="Green" title="E56">A1</font></i>: 
(  <a href="substut1.html#K37" title="SUBSTUT1:func.37">CQC_Sub</a> <span class="p1">(<span class="default"><a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">e</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">e</font></span>)</span> &amp; ( for <font color="Olive" title="b1">S'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a>  holds <br/> ( ( <font color="Olive" title="b1">S'</font> is <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a> implies <font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">S'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a>  ) &amp; ( <font color="Olive" title="b1">S'</font> is <a href="substut1.html#V4" title="SUBSTUT1:attr.4">Sub_atomic</a> implies <font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">S'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K15" title="QC_LANG1:func.15">the_pred_symbol_of</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">S'</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><a href="substut1.html#K26" title="SUBSTUT1:func.26">Sub_the_arguments_of</a> <font color="Olive" title="b1">S'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">S'</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span> ) &amp; ( <font color="Olive" title="b1">S'</font> is <a href="substut1.html#V5" title="SUBSTUT1:attr.5">Sub_negative</a> implies <font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">S'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Olive" title="b1">S'</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive" title="b1">S'</font> is <a href="substut1.html#V6" title="SUBSTUT1:attr.6">Sub_conjunctive</a> implies <font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">S'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K28" title="SUBSTUT1:func.28">Sub_the_left_argument_of</a> <font color="Olive" title="b1">S'</font></span>)</span></span>)</span> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K29" title="SUBSTUT1:func.29">Sub_the_right_argument_of</a> <font color="Olive" title="b1">S'</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive" title="b1">S'</font> is <a href="substut1.html#V7" title="SUBSTUT1:attr.7">Sub_universal</a> implies <font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">S'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K36" title="SUBSTUT1:func.36">Quant</a> <font color="Olive" title="b1">S'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="substut1.html#K31" title="SUBSTUT1:func.31">Sub_the_scope_of</a> <font color="Olive" title="b1">S'</font></span>)</span></span>)</span> ) ) ) )
 <b>by </b><i><a class="ref" href="substut1.html#D38" target="_self" title="SUBSTUT1:def.38">Def38</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">l</font> =  <a href="substut1.html#K26" title="SUBSTUT1:func.26">Sub_the_arguments_of</a> <span class="p1">(<span class="default"><a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">e</font></span>)</span>;<br/>
<a NAME="E3:92"/><i><font color="Green" title="E57">A2</font></i>: 
 <a href="substut1.html#K26" title="SUBSTUT1:func.26">Sub_the_arguments_of</a> <span class="p1">(<span class="default"><a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">e</font></span>)</span> is   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="ref" href="substut1.html#D29" target="_self" title="SUBSTUT1:def.29">Def29</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">l</font> =  <a href="substut1.html#K26" title="SUBSTUT1:func.26">Sub_the_arguments_of</a> <span class="p1">(<span class="default"><a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">e</font></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  <b>by </b><i><a class="ref" href="substut1.html#T34" target="_self" title="SUBSTUT1:th.34">Th34</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">s</font> =  <a href="substut1.html#K5" title="SUBSTUT1:func.5">CQC_Subst</a> <font color="Maroon" title="c7">l</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">e</font></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  ;<br/>
<a NAME="E6:92"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="txt" href="#E3:92"><i><font color="Green" title="E57">A2</font></i></a>, <a class="ref" href="qc_lang1.html#D8" title="QC_LANG1:def.8">QC_LANG1:def 8</a></i>;<br/>
<a NAME="E7:92"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="ref" href="substut1.html#D3" target="_self" title="SUBSTUT1:def.3">Def3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">s</font> = <font color="Maroon" title="c8">s</font> as   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</font> <b>by </b><i><a class="ref" href="substut1.html#T34" target="_self" title="SUBSTUT1:th.34">Th34</a></i>;<br/>
<a NAME="E9:92"/>
 <a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">e</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#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">ll</font></span>)</span>,<font color="Maroon" title="c4">e</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="ref" href="substut1.html#T9" target="_self" title="SUBSTUT1:th.9">Th9</a></i>;<br/>
<a NAME="E10:92"/><b>then </b>
<span class="p1">(<span class="default"><a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">e</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a>  <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> <font color="Maroon" title="c3">ll</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">P'</font> =  <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="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">e</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span> as   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c1">k</font> <b>by </b><i><a class="txt" href="substut1.html#E78"><i><font color="Green" title="E53">Lm6</font></i></a></i>;<br/>
<a NAME="E12:92"/>
 <a href="substut1.html#K37" title="SUBSTUT1:func.37">CQC_Sub</a> <span class="p1">(<span class="default"><a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">e</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">P'</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c9">s</font>
 
<b>by </b><i><a class="txt" href="#E2:92"><i><font color="Green" title="E56">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:92"/>
 <a href="substut1.html#K37" title="SUBSTUT1:func.37">CQC_Sub</a> <span class="p1">(<span class="default"><a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">e</font></span>)</span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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