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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <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> ] <span class="kw">means </span>( $1 is  <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a>  or $1 is  <a href="substut1.html#V4" title="SUBSTUT1:attr.4">Sub_atomic</a>  or $1 is  <a href="substut1.html#V5" title="SUBSTUT1:attr.5">Sub_negative</a>  or $1 is  <a href="substut1.html#V6" title="SUBSTUT1:attr.6">Sub_conjunctive</a>  or $1 is  <a href="substut1.html#V7" title="SUBSTUT1:attr.7">Sub_universal</a>  );<br/>
<a NAME="E1:51"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <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> <br/>  for <font color="Olive" title="b2">p</font> being   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Olive" title="b1">k</font><br/>  for <font color="Olive" title="b3">ll</font> being   <a href="qc_lang1.html#NM7" title="QC_LANG1:NM.7">QC-variable_list</a> of <font color="Olive" title="b1">k</font><br/>  for <font color="Olive" title="b4">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  <font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> (<font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">ll</font>,<font color="Olive" title="b4">e</font>)]
 
;<br/>
<a NAME="E2:51"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
 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>   st <font color="Olive" title="b1">S</font> is  <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">S</font>]
 
;<br/>
<a NAME="E3:51"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/>  for <font color="Olive" title="b2">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> <br/>  for <font color="Olive" title="b3">SQ</font> being    <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Olive" title="b2">S</font>,<font color="Olive" title="b1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>  st <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Olive" title="b2">S</font>,<font color="Olive" title="b1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">S</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> (<span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Olive" title="b2">S</font>,<font color="Olive" title="b1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>,<font color="Olive" title="b3">SQ</font>)]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="substut1.html#D28" target="_self" title="SUBSTUT1:def.28">Def28</a></span>;<br/>
<a NAME="E4:51"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
 for <font color="Olive" title="b1">S1</font>, <font color="Olive" title="b2">S2</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>   st <font color="Olive" title="b1">S1</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">S2</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">S1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">S2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K21" title="SUBSTUT1:func.21">Sub_&amp;</a> (<font color="Olive" title="b1">S1</font>,<font color="Olive" title="b2">S2</font>)]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="substut1.html#D27" target="_self" title="SUBSTUT1:def.27">Def27</a></span>;<br/>
<a NAME="E5:51"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
 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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">S</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K20" title="SUBSTUT1:func.20">Sub_not</a> <font color="Olive" title="b1">S</font>]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="substut1.html#D26" target="_self" title="SUBSTUT1:def.26">Def26</a></span>;<br/>
<span class="kw">thus </span><a NAME="E6:51"/>
 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  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">S</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="substut1.html#S1" target="_self" title="SUBSTUT1:sch.1">SUBSTUT1:sch 1</a>(<a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E5:51"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E29">A3</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/>


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