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<a NAME="E1:67_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">P</font> being  <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Olive">k</font><br/> for <font color="Olive">ll</font> being   <a href="qc_lang1.html#M1">QC-variable_list</a> of <font color="Olive">k</font><br/> for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K1" target="_self">vSUB</a>  holds  <font color="Maroon">P<sub>1</sub></font>[ <a href="substut1.html#K17" target="_self">Sub_P</a> <font color="Olive">P</font>,<font color="Olive">ll</font>,<font color="Olive">e</font>]
 
<b>by </b><i><a class="ref" href="substut1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E2:67_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">S</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>   st <font color="Olive">S</font> is <a href="substut1.html#V2" target="_self">Sub_VERUM</a> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">S</font>]
 
<b>by </b><i><a class="ref" href="substut1.html#D3" target="_self">Def3</a></i>;<br/>
<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">S</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ;<br/><b>assume </b><a NAME="E1:67_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">S</font>]
 ;<br/><a NAME="E2:67_1_1"/>
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K27" target="_self">Sub_the_argument_of</a> <span class="p1">(<span class="default"><a href="substut1.html#K20" target="_self">Sub_not</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:67_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="substut1.html#K20" target="_self">Sub_not</a> <font color="Maroon">S</font>]
 <b>by </b><i><a class="ref" href="substut1.html#D3" target="_self">Def3</a>, <a class="txt" href="substut1.html#E1:67_1"><i><font color="Green">E43</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">S1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> , <font color="Maroon">S2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ;<br/><b>assume </b><a NAME="E1:67_1_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">S1</font> <a href="substut1.html#K19" target="_self">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="substut1.html#K19" target="_self">`2</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">S1</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">S2</font>] )
 ;<br/><a NAME="E2:67_1_2"/><i><font color="Green">E50</font></i>: 
 <a href="substut1.html#K21" target="_self">Sub_&amp;</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font> is <a href="substut1.html#V6" target="_self">Sub_conjunctive</a>
 <b>by </b><i>, </i>;<br/><a NAME="E3:67_1_2"/>
( <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <span class="p1">(<span class="default"><a href="substut1.html#K21" target="_self">Sub_&amp;</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span> &amp; <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <span class="p1">(<span class="default"><a href="substut1.html#K21" target="_self">Sub_&amp;</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:67_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="substut1.html#K21" target="_self">Sub_&amp;</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font>]
 <b>by </b><i><a class="ref" href="substut1.html#D3" target="_self">Def3</a>, , <a class="txt" href="substut1.html#E2:67_1"><i><font color="Green">E45</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">S</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ;<br/><b>let </b><font color="Maroon">SQ</font> be    <a href="substut1.html#M1" target="_self">second_Q_comp</a> of <span class="p1"><a href="substut1.html#K25" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="substut1.html#K25" target="_self">]</a></span>;<br/><b>assume </b><a NAME="E1:67_1_3"/><i><font color="Green">E52</font></i>: 
( <span class="p1"><a href="substut1.html#K25" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="substut1.html#K25" target="_self">]</a></span> is <a href="substut1.html#V3" target="_self">quantifiable</a> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">S</font>] )
 ;<br/><a NAME="E2:67_1_3"/><i><font color="Green">E53</font></i>: 
 <a href="substut1.html#K24" target="_self">Sub_All</a> <span class="p1"><a href="substut1.html#K25" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="substut1.html#K25" target="_self">]</a></span>,<font color="Maroon">SQ</font> is <a href="substut1.html#V7" target="_self">Sub_universal</a>
 <b>by </b><i><a class="txt" href="substut1.html#E2:67_1"><i><font color="Green">E45</font></i></a>, </i>;<br/><a NAME="E3:67_1_3"/>
<span class="p1"><a href="substut1.html#K25" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="substut1.html#K25" target="_self">]</a></span> <a href="substut1.html#K22" target="_self">`1</a>  <a href="hidden.html#R1">=</a>  <a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <span class="p1">(<span class="default"><a href="substut1.html#K24" target="_self">Sub_All</a> <span class="p2"><a href="substut1.html#K25" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="substut1.html#K25" target="_self">]</a></span>,<font color="Maroon">SQ</font></span>)</span>
 <b>by </b><i><a class="txt" href="substut1.html#E2:67_1"><i><font color="Green">E45</font></i></a>, </i>;<br/><a NAME="E4:67_1_3"/><b>then </b>
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K31" target="_self">Sub_the_scope_of</a> <span class="p1">(<span class="default"><a href="substut1.html#K24" target="_self">Sub_All</a> <span class="p2"><a href="substut1.html#K25" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="substut1.html#K25" target="_self">]</a></span>,<font color="Maroon">SQ</font></span>)</span>
 <b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><b>hence </b><a NAME="E5:67_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="substut1.html#K24" target="_self">Sub_All</a> <span class="p1"><a href="substut1.html#K25" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="substut1.html#K25" target="_self">]</a></span>,<font color="Maroon">SQ</font>]
 <b>by </b><i><a class="ref" href="substut1.html#D3" target="_self">Def3</a>, <a class="txt" href="substut1.html#E2:67_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="substut1.html#E3:67_1"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E6:67_1"/>
for <font color="Olive">S</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>  holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">S</font>]
 <b>from </b><i><a class="ref" href="substut1.html#S1" target="_self">SUBSTUT1:sch 1</a>(<a class="txt" href="substut1.html#E1:67"><i><font color="Green">E40</font></i></a>, , , <a class="txt" href="substut1.html#E1:67_1"><i><font color="Green">E43</font></i></a>, );</i><br/>


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