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<span class="kw">let </span><font color="Maroon" title="c2">ll1</font>, <font color="Maroon" title="c3">ll2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="qc_lang1.html#K1" title="QC_LANG1:func.1">QC-variables</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <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>  ex <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> ex <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> ex <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>  st <br/>( <font color="Maroon" title="c2">ll1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">ll</font> &amp; <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> ) &amp;  ex <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>  ex <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> ex <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> ex <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>  st <br/>( <font color="Maroon" title="c3">ll2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">ll</font> &amp; <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> ) implies <font color="Maroon" title="c2">ll1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">ll2</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c4">k1</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> , <font color="Maroon" title="c5">P1</font> being   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c4">k1</font>, <font color="Maroon" title="c6">ll19</font> being   <a href="qc_lang1.html#NM7" title="QC_LANG1:NM.7">QC-variable_list</a> of <font color="Maroon" title="c4">k1</font>, <font color="Maroon" title="c7">e1</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> <span class="kw"> such that </span><a NAME="E2:52_1_2"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
<font color="Maroon" title="c2">ll1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">ll19</font>
 <span class="kw">and </span><br/><a NAME="E3:52_1_2"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">ll19</font>,<font color="Maroon" title="c7">e1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 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 <br/>(  not <font color="Maroon" title="c3">ll2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">ll</font> or  not <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> ) or <font color="Maroon" title="c2">ll1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">ll2</font> )</span><br/>

<a NAME="E4:52_1_2"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
<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"><font color="Maroon" title="c5">P1</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c6">ll19</font></span>)</span>,<font color="Maroon" title="c7">e1</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="#E3:52_1_2"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="substut1.html#T9" target="_self" title="SUBSTUT1:th.9">Th9</a></span>;<br/>
<span class="kw">given </span><font color="Maroon" title="c8">k2</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> , <font color="Maroon" title="c9">P2</font> being   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c8">k2</font>, <font color="Maroon" title="c10">ll29</font> being   <a href="qc_lang1.html#NM7" title="QC_LANG1:NM.7">QC-variable_list</a> of <font color="Maroon" title="c8">k2</font>, <font color="Maroon" title="c11">e2</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> <span class="kw"> such that </span><a NAME="E6:52_1_2"/><span class="lab"><font color="Green" title="E31">A6</font></span>: 
<font color="Maroon" title="c3">ll2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c10">ll29</font>
 <span class="kw">and </span><br/><a NAME="E7:52_1_2"/><span class="lab"><font color="Green" title="E32">A7</font></span>: 
<font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c9">P2</font>,<font color="Maroon" title="c10">ll29</font>,<font color="Maroon" title="c11">e2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">ll1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">ll2</font></span><br/>

<a NAME="E8:52_1_2"/>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">P1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">ll19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P1</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c6">ll19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#T23" title="QC_LANG1:th.23">QC_LANG1:23</a></span>;<br/>
<a NAME="E9:52_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A8</font></span>: 
( <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">P2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c10">ll29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">P2</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c10">ll29</font> &amp; <font color="Maroon" title="c1">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">P1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">ll19</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52_1_2"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a>, <a class="ref" href="qc_lang1.html#T23" title="QC_LANG1:th.23">QC_LANG1:23</a></span>;<br/>
<a NAME="E10:52_1_2"/><span class="lab"><font color="Green" title="E34">A9</font></span>: 
<font color="Maroon" title="c1">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a>  is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="substut1.html#T11" target="_self" title="SUBSTUT1:th.11">Th11</a></span>;<br/>
<a NAME="E11:52_1_2"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
<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"><font color="Maroon" title="c9">P2</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c10">ll29</font></span>)</span>,<font color="Maroon" title="c11">e2</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="#E7:52_1_2"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="ref" href="substut1.html#T9" target="_self" title="SUBSTUT1:th.9">Th9</a></span>;<br/>
<a NAME="E12:52_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A11</font></span>: 
<font color="Maroon" title="c1">S</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="c9">P2</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c10">ll29</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E13:52_1_2"/>
<font color="Maroon" title="c1">S</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="c5">P1</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c6">ll19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52_1_2"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c5">P1</font> = 
 <a href="qc_lang1.html#K15" title="QC_LANG1:func.15">the_pred_symbol_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <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="#E10:52_1_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="ref" href="qc_lang1.html#D21" title="QC_LANG1:def.21">QC_LANG1:def 21</a></span>
<br/>.= 
<font color="Maroon" title="c9">P2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:52_1_2"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="txt" href="#E10:52_1_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="ref" href="qc_lang1.html#D21" title="QC_LANG1:def.21">QC_LANG1:def 21</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E15:52_1_2"/>
<font color="Maroon" title="c2">ll1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">ll2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52_1_2"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E6:52_1_2"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="txt" href="#E11:52_1_2"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="txt" href="#E9:52_1_2"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="finseq_1.html#T46" title="FINSEQ_1:th.46">FINSEQ_1:46</a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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