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<span class="kw">let </span><font color="Maroon" title="c1">B1</font>, <font color="Maroon" title="c2">B2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>; <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">SQ1</font> being    <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <font color="Maroon" title="c1">B1</font><br/>  for <font color="Olive" title="b2">SQ2</font> being    <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <font color="Maroon" title="c2">B2</font>  st <font color="Maroon" title="c1">B1</font> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>  &amp; <font color="Maroon" title="c2">B2</font> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>  &amp;  <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> (<font color="Maroon" title="c1">B1</font>,<font color="Olive" title="b1">SQ1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> (<font color="Maroon" title="c2">B2</font>,<font color="Olive" title="b2">SQ2</font>) holds <br/><font color="Maroon" title="c1">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">SQ1</font> be    <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <font color="Maroon" title="c1">B1</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">SQ2</font> being    <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <font color="Maroon" title="c2">B2</font>  st <font color="Maroon" title="c1">B1</font> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>  &amp; <font color="Maroon" title="c2">B2</font> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>  &amp;  <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> (<font color="Maroon" title="c1">B1</font>,<font color="Maroon" title="c3">SQ1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> (<font color="Maroon" title="c2">B2</font>,<font color="Olive" title="b1">SQ2</font>) holds <br/><font color="Maroon" title="c1">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">SQ2</font> be    <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <font color="Maroon" title="c2">B2</font>; <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">B1</font> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>  &amp; <font color="Maroon" title="c2">B2</font> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>  &amp;  <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> (<font color="Maroon" title="c1">B1</font>,<font color="Maroon" title="c3">SQ1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> (<font color="Maroon" title="c2">B2</font>,<font color="Maroon" title="c4">SQ2</font>) implies <font color="Maroon" title="c1">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B2</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:65"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<font color="Maroon" title="c1">B1</font> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a> 
 <span class="kw">and </span><br/><a NAME="E2:65"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<font color="Maroon" title="c2">B2</font> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a> 
 <span class="kw">and </span><br/><a NAME="E3:65"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
 <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> (<font color="Maroon" title="c1">B1</font>,<font color="Maroon" title="c3">SQ1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> (<font color="Maroon" title="c2">B2</font>,<font color="Maroon" title="c4">SQ2</font>)
 ; <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">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B2</font></span><br/>

<a NAME="E4:65"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
(  <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> (<font color="Maroon" title="c1">B1</font>,<font color="Maroon" title="c3">SQ1</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"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="substut1.html#K23" title="SUBSTUT1:func.23">`2</a></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="substut1.html#K22" title="SUBSTUT1:func.22">`1</a></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a></span>)</span>)</span>)</span>,<font color="Maroon" title="c3">SQ1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp;  <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> (<font color="Maroon" title="c2">B2</font>,<font color="Maroon" title="c4">SQ2</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"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="substut1.html#K23" title="SUBSTUT1:func.23">`2</a></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="substut1.html#K22" title="SUBSTUT1:func.22">`1</a></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a></span>)</span>)</span>)</span>,<font color="Maroon" title="c4">SQ2</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="txt" href="#E1:65"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="substut1.html#D24" target="_self" title="SUBSTUT1:def.24">Def24</a></span>;<br/>
<a NAME="E5:65"/><span class="kw">then </span>
 <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="substut1.html#K23" title="SUBSTUT1:func.23">`2</a></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="substut1.html#K22" title="SUBSTUT1:func.22">`1</a></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="substut1.html#K23" title="SUBSTUT1:func.23">`2</a></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="substut1.html#K22" title="SUBSTUT1:func.22">`1</a></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E6:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A5</font></span>: 
( <font color="Maroon" title="c1">B1</font> <a href="substut1.html#K23" title="SUBSTUT1:func.23">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B2</font> <a href="substut1.html#K23" title="SUBSTUT1:func.23">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="substut1.html#K22" title="SUBSTUT1:func.22">`1</a></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="substut1.html#K22" title="SUBSTUT1:func.22">`1</a></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#T5" title="QC_LANG2:th.5">QC_LANG2:5</a></span>;<br/>
<a NAME="E7:65"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a>  &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  &amp; <font color="Maroon" title="c2">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E8:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A6</font></span>: 
<font color="Maroon" title="c2">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="substut1.html#K22" title="SUBSTUT1:func.22">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="substut1.html#K23" title="SUBSTUT1:func.23">`2</a></span>)</span></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T8" title="MCART_1:th.8">MCART_1:8</a></span>;<br/>
<a NAME="E9:65"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a>  &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  &amp; <font color="Maroon" title="c1">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E10:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A7</font></span>: 
<font color="Maroon" title="c1">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="substut1.html#K22" title="SUBSTUT1:func.22">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="substut1.html#K23" title="SUBSTUT1:func.23">`2</a></span>)</span></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T8" title="MCART_1:th.8">MCART_1:8</a></span>;<br/>
<a NAME="E11:65"/><span class="lab"><font color="Green" title="E43">A8</font></span>: 
( <font color="Maroon" title="c1">B1</font> <a href="substut1.html#K22" title="SUBSTUT1:func.22">`1</a>  <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"><span class="p3">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="substut1.html#K22" title="SUBSTUT1:func.22">`1</a></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="substut1.html#K22" title="SUBSTUT1:func.22">`1</a></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c2">B2</font> <a href="substut1.html#K22" title="SUBSTUT1:func.22">`1</a>  <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"><span class="p3">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="substut1.html#K22" title="SUBSTUT1:func.22">`1</a></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="substut1.html#K22" title="SUBSTUT1:func.22">`1</a></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a></span>)</span></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#T10" target="_self" title="SUBSTUT1:th.10">Th10</a></span>;<br/>
<a NAME="E12:65"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="substut1.html#K22" title="SUBSTUT1:func.22">`1</a></span>)</span> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="substut1.html#K15" title="SUBSTUT1:func.15">QSub</a> <a href="funct_1.html#K1" title="FUNCT_1:func.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"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="substut1.html#K23" title="SUBSTUT1:func.23">`2</a></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="substut1.html#K22" title="SUBSTUT1:func.22">`1</a></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a></span>)</span>)</span>)</span>,<font color="Maroon" title="c3">SQ1</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="txt" href="#E1:65"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="substut1.html#D23" target="_self" title="SUBSTUT1:def.23">Def23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:65"/>
<font color="Maroon" title="c1">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E4:65"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E6:65"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="txt" href="#E11:65"><span class="lab"><font color="Green" title="E43">A8</font></span></a>, <a class="txt" href="#E10:65"><span class="lab"><font color="Green" title="E42">A7</font></span></a>, <a class="txt" href="#E8:65"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="ref" href="substut1.html#D23" target="_self" title="SUBSTUT1:def.23">Def23</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/>


</div>
