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<span class="kw">let </span><font color="Maroon" title="c1">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"> ( <font color="Maroon" title="c1">S</font> is  <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a>  implies  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">VERUM</a>  )</span><br/>

<a NAME="E1:8"/>
 ex <font color="Olive" title="b1">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> st <br/>(  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">S</font> &amp; (  for <font color="Olive" title="b2">S9</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="b2">S9</font> is  <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a>  implies <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">S9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">VERUM</a>  ) &amp; ( <font color="Olive" title="b2">S9</font> is  <a href="substut1.html#V4" title="SUBSTUT1:attr.4">Sub_atomic</a>  implies <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">S9</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="b2">S9</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="b2">S9</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">S9</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a></span>)</span>)</span>)</span> ) &amp; ( <font color="Olive" title="b2">S9</font> is  <a href="substut1.html#V5" title="SUBSTUT1:attr.5">Sub_negative</a>  implies <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">S9</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="Olive" title="b1">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="b2">S9</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive" title="b2">S9</font> is  <a href="substut1.html#V6" title="SUBSTUT1:attr.6">Sub_conjunctive</a>  implies <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">S9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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="b2">S9</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="Olive" title="b1">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="b2">S9</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive" title="b2">S9</font> is  <a href="substut1.html#V7" title="SUBSTUT1:attr.7">Sub_universal</a>  implies <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">S9</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="b2">S9</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">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="b2">S9</font></span>)</span></span>)</span>) ) ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="substut1.html#D38" title="SUBSTUT1:def.38">SUBSTUT1:def 38</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:8"/>
( <font color="Maroon" title="c1">S</font> is  <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a>  implies  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">VERUM</a>  )
 ; <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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