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<b>let </b><font color="Maroon" title="c2">S1</font>, <font color="Maroon" title="c3">S2</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <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 <br/>( <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K21" title="SUBSTUT1:func.21">Sub_&amp;</a> <font color="Maroon" title="c2">S1</font>,<font color="Olive" title="b1">S'</font> &amp; <font color="Maroon" title="c2">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="b1">S'</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  ) &amp;  ex <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 <br/>( <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K21" title="SUBSTUT1:func.21">Sub_&amp;</a> <font color="Maroon" title="c3">S2</font>,<font color="Olive" title="b1">S'</font> &amp; <font color="Maroon" title="c3">S2</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="b1">S'</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  ) implies <font color="Maroon" title="c2">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font> )</span><br/>



<b>given </b><font color="Maroon" title="c4">T1</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> <b> such that </b><a NAME="E2:54_1_1"/><i><font color="Green" title="E30">A2</font></i>: 
( <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K21" title="SUBSTUT1:func.21">Sub_&amp;</a> <font color="Maroon" title="c2">S1</font>,<font color="Maroon" title="c4">T1</font> &amp; <font color="Maroon" title="c2">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="Maroon" title="c4">T1</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</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">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/>(  not <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K21" title="SUBSTUT1:func.21">Sub_&amp;</a> <font color="Maroon" title="c3">S2</font>,<font color="Olive" title="b1">S'</font> or  not <font color="Maroon" title="c3">S2</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="b1">S'</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  ) or <font color="Maroon" title="c2">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font> )</span><br/>

<b>given </b><font color="Maroon" title="c5">T2</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> <b> such that </b><a NAME="E4:54_1_1"/><i><font color="Green" title="E31">A3</font></i>: 
( <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K21" title="SUBSTUT1:func.21">Sub_&amp;</a> <font color="Maroon" title="c3">S2</font>,<font color="Maroon" title="c5">T2</font> &amp; <font color="Maroon" title="c3">S2</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T2</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font></span><br/>

<a NAME="E5:54_1_1"/><i><font color="Green" title="E32">A4</font></i>: 
( <font color="Maroon" title="c1">S</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"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">S1</font> <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="c1">S</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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">T2</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">S2</font> <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> )
 
<b>by </b><i><a class="txt" href="substut1.html#E2:54_1_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="substut1.html#E4:54_1_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="substut1.html#D21" target="_self" title="SUBSTUT1:def.21">Def21</a></i>;<br/>
<a NAME="E6:54_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </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="c4">T1</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </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">T2</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>then </b><span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </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">T2</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">T2</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></i>
;<br/>
<a NAME="E8:54_1_1"/><b>then </b><i><font color="Green" title="E33">A5</font></i>: 
<span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">T2</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T46" title="FINSEQ_1:th.46">FINSEQ_1:46</a></i>;<br/>
<a NAME="E9:54_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> )
 
;<br/>
<b>then consider </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">c</font>, <font color="Maroon" title="c9">d</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E11:54_1_1"/><i><font color="Green" title="E34">A6</font></i>: 
( ( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span> ) or ( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span> ) )
 <b>and </b><br/><a NAME="E12:54_1_1"/><i><font color="Green" title="E35">A7</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">b</font>
 <b>and </b><br/><a NAME="E13:54_1_1"/><i><font color="Green" title="E36">A8</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c9">d</font>
 <b>by </b><i><a class="txt" href="substut1.html#E8:54_1_1"><i><font color="Green" title="E33">A5</font></i></a></i>;<br/>
<a NAME="E14:54_1_1"/>
 ex <font color="Olive" title="b1">t</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <font color="Maroon" title="c6">a</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font>
 
<b>by </b><i><a class="txt" href="substut1.html#E12:54_1_1"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="substut1.html#E13:54_1_1"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="finseq_1.html#T64" title="FINSEQ_1:th.64">FINSEQ_1:64</a></i>;<br/>
<a NAME="E15:54_1_1"/><b>then </b><i><font color="Green" title="E37">A9</font></i>: 
<font color="Maroon" title="c2">S1</font> <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="c3">S2</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> 
 
<b>by </b><i><a class="txt" href="substut1.html#E11:54_1_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="qc_lang1.html#T37" title="QC_LANG1:th.37">QC_LANG1:37</a></i>;<br/>
<a NAME="E16:54_1_1"/>
( <font color="Maroon" title="c2">S1</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">S1</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">S1</font> <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="c3">S2</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="c3">S2</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">S2</font> <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> )
 
<b>by </b><i><a class="ref" href="substut1.html#T10" target="_self" title="SUBSTUT1:th.10">Th10</a></i>;<br/>
<b>hence </b><a NAME="E17:54_1_1"/>
<font color="Maroon" title="c2">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font>
 <b>by </b><i><a class="txt" href="substut1.html#E5:54_1_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="substut1.html#E15:54_1_1"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>; <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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