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<span class="kw">let </span><font color="Maroon" title="c3">ll1</font>, <font color="Maroon" title="c4">ll2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">QC-variables</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b2">P</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Olive" title="b1">k</font>,<font color="Maroon" title="c1">A</font> ex <font color="Olive" title="b3">ll</font> being   <a href="qc_lang1.html#NM9" title="QC_LANG1:NM.9">QC-variable_list</a> of <font color="Olive" title="b1">k</font>,<font color="Maroon" title="c1">A</font> st <br/>( <font color="Maroon" title="c3">ll1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">ll</font> &amp; <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">P</font> <a href="qc_lang1.html#K10" title="QC_LANG1:func.10">!</a> <font color="Olive" title="b3">ll</font> ) &amp;  ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b2">P</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Olive" title="b1">k</font>,<font color="Maroon" title="c1">A</font> ex <font color="Olive" title="b3">ll</font> being   <a href="qc_lang1.html#NM9" title="QC_LANG1:NM.9">QC-variable_list</a> of <font color="Olive" title="b1">k</font>,<font color="Maroon" title="c1">A</font> st <br/>( <font color="Maroon" title="c4">ll2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">ll</font> &amp; <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">P</font> <a href="qc_lang1.html#K10" title="QC_LANG1:func.10">!</a> <font color="Olive" title="b3">ll</font> ) implies <font color="Maroon" title="c3">ll1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ll2</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c5">k1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>, <font color="Maroon" title="c6">P1</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Maroon" title="c5">k1</font>,<font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c7">ll19</font> being   <a href="qc_lang1.html#NM9" title="QC_LANG1:NM.9">QC-variable_list</a> of <font color="Maroon" title="c5">k1</font>,<font color="Maroon" title="c1">A</font><span class="kw"> such that </span><a NAME="E2:45_1_1"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
<font color="Maroon" title="c3">ll1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">ll19</font>
 <span class="kw">and </span><br/><a NAME="E3:45_1_1"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P1</font> <a href="qc_lang1.html#K10" title="QC_LANG1:func.10">!</a> <font color="Maroon" title="c7">ll19</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">P</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Olive" title="b1">k</font>,<font color="Maroon" title="c1">A</font><br/>  for <font color="Olive" title="b3">ll</font> being   <a href="qc_lang1.html#NM9" title="QC_LANG1:NM.9">QC-variable_list</a> of <font color="Olive" title="b1">k</font>,<font color="Maroon" title="c1">A</font>  holds <br/>(  not <font color="Maroon" title="c4">ll2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">ll</font> or  not <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">P</font> <a href="qc_lang1.html#K10" title="QC_LANG1:func.10">!</a> <font color="Olive" title="b3">ll</font> ) or <font color="Maroon" title="c3">ll1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ll2</font> )</span><br/>

<a NAME="E4:45_1_1"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<font color="Maroon" title="c2">F</font> <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="c6">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="c7">ll19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_1_1"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="qc_lang1.html#T8" target="_self" onmouseover="rs('qc_lang1/T8')" onmouseout="rh()">Th8</a></span>;<br/>
<span class="kw">given </span><font color="Maroon" title="c8">k2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>, <font color="Maroon" title="c9">P2</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Maroon" title="c8">k2</font>,<font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c10">ll29</font> being   <a href="qc_lang1.html#NM9" title="QC_LANG1:NM.9">QC-variable_list</a> of <font color="Maroon" title="c8">k2</font>,<font color="Maroon" title="c1">A</font><span class="kw"> such that </span><a NAME="E6:45_1_1"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
<font color="Maroon" title="c4">ll2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">ll29</font>
 <span class="kw">and </span><br/><a NAME="E7:45_1_1"/><span class="lab"><font color="Green" title="E30">A6</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">P2</font> <a href="qc_lang1.html#K10" title="QC_LANG1:func.10">!</a> <font color="Maroon" title="c10">ll29</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">ll1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ll2</font></span><br/>

<a NAME="E8:45_1_1"/><span class="lab"><font color="Green" title="E31">A7</font></span>: 
<font color="Maroon" title="c2">F</font> <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="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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="qc_lang1.html#T8" target="_self" onmouseover="rs('qc_lang1/T8')" onmouseout="rh()">Th8</a></span>;<br/>
<font color="Maroon" title="c6">P1</font> = 
 <a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_pred_symbol_of</a> <font color="Maroon" title="c2">F</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E3:45_1_1"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="qc_lang1.html#D22" target="_self" onmouseover="rs('qc_lang1/D22')" onmouseout="rh()">Def22</a></span>
<br/>.= 
<font color="Maroon" title="c9">P2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E7:45_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="qc_lang1.html#D22" target="_self" onmouseover="rs('qc_lang1/D22')" onmouseout="rh()">Def22</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:45_1_1"/>
<font color="Maroon" title="c3">ll1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ll2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1_1"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E6:45_1_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E4:45_1_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E8:45_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="finseq_1.html#T33" onmouseover="rs('finseq_1/T33')" onmouseout="rh()">FINSEQ_1:33</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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