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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon">F<sub>2</sub></font>(),  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>( $1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() &amp; (  for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $2 holds <br/>( ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies $1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">p</font>) ) &amp; ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  implies $1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default">$1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies $1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default">$1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default">$1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies $1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive" title="b1">p</font>,<span class="p1">(<span class="default">$1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>) ) ) ) );<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>(),  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon">F<sub>2</sub></font>(),   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()] <span class="kw">means </span>( ( $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>() implies $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() ) &amp; ( $3 is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>($3) ) &amp; ( $3 is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  implies $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default">$2 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> $3</span>)</span></span>)</span>) ) &amp; ( $3 is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default">$2 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> $3</span>)</span></span>)</span>,<span class="p1">(<span class="default">$2 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> $3</span>)</span></span>)</span>) ) &amp; ( $3 is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>($3,<span class="p1">(<span class="default">$2 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> $3</span>)</span></span>)</span>) ) );<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> ex <font color="Olive" title="b1">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">F</font>,$1];<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; (  for <font color="Olive" title="b2">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon">F<sub>2</sub></font>()  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">g</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Olive" title="b1">p</font></span>)</span>] holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> ) );<br/>
<a NAME="E1:58_1"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <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">S<sub>3</sub></font>[<font color="Maroon" title="c1">n</font>] implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c2">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon">F<sub>2</sub></font>()<span class="kw"> such that </span><a NAME="E2:58_1_1"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">n</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">S<sub>3</sub></font>[<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>5</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>(),   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()] <span class="kw">means </span>( (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> $1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 ) &amp; (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <font color="Maroon">S<sub>2</sub></font>[$2,<font color="Maroon" title="c2">F</font>,$1] ) );<br/>
<a NAME="E3:58_1_1"/><span class="lab"><font color="Green" title="E40">A3</font></span>: 
 for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()  ex <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">p</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></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">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1"><span class="kw">now </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"> ( (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> ) 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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>() &amp;  ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">p</font>] ) 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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  &amp;  ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">p</font>] ) 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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  &amp;  ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">p</font>] ) 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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  &amp;  ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">p</font>] ) 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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  &amp;  ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">p</font>] ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:58_1_1_1_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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>() ) 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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  ) 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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  ) 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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  ) 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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  ) )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#T9" target="_self" onmouseover="rs('qc_lang1/T9')" onmouseout="rh()">Th9</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:58_1_1_1_1_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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font></span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c4">y</font> = <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</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="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font></span><br/><span class="kw">thus </span><a NAME="E2:58_1_1_1_1_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:58_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></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">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">p</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c4">y</font> = <font color="Maroon">F<sub>3</sub></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">S<sub>2</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">p</font>]</span><br/><span class="kw">thus </span><a NAME="E2:58_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">p</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="ref" href="qc_lang1.html#T20" target="_self" onmouseover="rs('qc_lang1/T20')" onmouseout="rh()">Th20</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_3"><span class="kw">case </span></a><a NAME="E1:58_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E41">A5</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  )
 ; <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">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">p</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c4">y</font> = <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c3">p</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">S<sub>2</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">p</font>]</span><br/><span class="kw">thus </span><a NAME="E2:58_1_1_1_1_1_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">p</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="ref" href="qc_lang1.html#T20" target="_self" onmouseover="rs('qc_lang1/T20')" onmouseout="rh()">Th20</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_4"><span class="kw">case </span></a><a NAME="E1:58_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E41">A6</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  )
 ; <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">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">p</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c4">y</font> = <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</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"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">p</font>]</span><br/><span class="kw">thus </span><a NAME="E2:58_1_1_1_1_1_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">p</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="ref" href="qc_lang1.html#T20" target="_self" onmouseover="rs('qc_lang1/T20')" onmouseout="rh()">Th20</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_5"><span class="kw">case </span></a><a NAME="E1:58_1_1_1_1_1_5"/><span class="lab"><font color="Green" title="E41">A7</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  )
 ; <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">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">p</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c4">y</font> = <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</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"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">p</font>]</span><br/><span class="kw">thus </span><a NAME="E2:58_1_1_1_1_1_5"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">p</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_1_1_1_5"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="ref" href="qc_lang1.html#T20" target="_self" onmouseover="rs('qc_lang1/T20')" onmouseout="rh()">Th20</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_6"><span class="kw">case </span></a><a NAME="E1:58_1_1_1_1_1_6"/><span class="lab"><font color="Green" title="E41">A8</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  )
 ; <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">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">p</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c4">y</font> = <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</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"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">p</font>]</span><br/><span class="kw">thus </span><a NAME="E2:58_1_1_1_1_1_6"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">p</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_1_1_1_6"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="ref" href="qc_lang1.html#T20" target="_self" onmouseover="rs('qc_lang1/T20')" onmouseout="rh()">Th20</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:58_1_1_1"/>
 ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">y</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c3">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon">F<sub>2</sub></font>()<span class="kw"> such that </span><br/><a NAME="E5:58_1_1"/><span class="lab"><font color="Green" title="E41">A9</font></span>: 
 for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c3">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E3:58_1_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>);<br/></span>
<span class="kw">take </span><font color="Maroon" title="c4">H</font> = <font color="Maroon" title="c3">G</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">S<sub>1</sub></font>[<font color="Maroon" title="c4">H</font>,<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">thus </span><a NAME="E6:58_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H</font>,<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
  <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><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:58_1_1_2"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></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">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/>( ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">p</font>) ) &amp; ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive" title="b1">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>) ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:58_1_1_2_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#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 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#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_2_1_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#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>()</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:58_1_1_2_1_1_1"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E41">A9</font></span></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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>()</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:58_1_1_2_1_1_2"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E41">A9</font></span></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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c5">p</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></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"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies ( ( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c5">p</font>) ) &amp; ( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:58_1_1_2"/><span class="lab"><font color="Green" title="E42">A10</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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="c5">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c5">p</font>) ) &amp; ( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>) ) )</span><br/>

<span class="kw">thus </span><a NAME="E3:58_1_1_2"/>
( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c5">p</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="c5">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>) ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_1_2_2_1"><span class="kw">now </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"> ( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c5">p</font>) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:58_1_1_2_2_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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_2_2_1_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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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="c5">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c5">p</font>) )</span><br/><div class="add"><a NAME="E2:58_1_1_2_2_1_1_1"/><span class="kw">then </span>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E2:58_1_1_2"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="ref" href="nat_1.html#T8" onmouseover="rs('nat_1/T8')" onmouseout="rh()">NAT_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:58_1_1_2_2_1_1_1"/>
( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c5">p</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1"><span class="lab"><font color="Green" title="E39">A2</font></span></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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_2_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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="c5">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c5">p</font>) )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:58_1_1_2_2_1_1_2"/>
( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c5">p</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E41">A9</font></span></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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:58_1_1_2_2"/>
( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c5">p</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"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E4:58_1_1_2"/>
( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</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"> ( ( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>) ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:58_1_1_2_3"/><span class="lab"><font color="Green" title="E43">A11</font></span>: 
<font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> 
 ; <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="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>)</span><br/>

<a NAME="E2:58_1_1_2_3"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1_2"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="ref" href="qc_lang1.html#T14" target="_self" onmouseover="rs('qc_lang1/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<a NAME="E3:58_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A12</font></span>: 
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E41">A9</font></span></a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_1_2_3_1"><span class="kw">now </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"> <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_3_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:58_1_1_2_3_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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_2_3_1_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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>)</span><br/><div class="add"><a NAME="E2:58_1_1_2_3_1_1_1"/><span class="kw">then </span>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E2:58_1_1_2"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="ref" href="nat_1.html#T8" onmouseover="rs('nat_1/T8')" onmouseout="rh()">NAT_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:58_1_1_2_3_1_1_1"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E1:58_1_1_2_3"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E3:58_1_1_2_3"><span class="lab"><font color="Green" title="E44">A12</font></span></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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_2_3_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:58_1_1_2_3_1_1_2"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E1:58_1_1_2_3"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E3:58_1_1_2_3"><span class="lab"><font color="Green" title="E44">A12</font></span></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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:58_1_1_2_3"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</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/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E5:58_1_1_2"/>
( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</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"> ( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:58_1_1_2_4"/><span class="lab"><font color="Green" title="E43">A13</font></span>: 
<font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> 
 ; <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="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>)</span><br/>

<a NAME="E2:58_1_1_2_4"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1_2"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="ref" href="qc_lang1.html#T15" target="_self" onmouseover="rs('qc_lang1/T15')" onmouseout="rh()">Th15</a></span>;<br/>
<a NAME="E3:58_1_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A14</font></span>: 
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E41">A9</font></span></a></span>;<br/>
<a NAME="E4:58_1_1_2_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1_2"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="txt" href="#E1:58_1_1_2_4"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="ref" href="qc_lang1.html#T15" target="_self" onmouseover="rs('qc_lang1/T15')" onmouseout="rh()">Th15</a></span>;<br/>
<a NAME="E5:58_1_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A15</font></span>: 
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E41">A9</font></span></a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_1_2_4_1"><span class="kw">now </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"> <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_4_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:58_1_1_2_4_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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_2_4_1_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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>)</span><br/><div class="add"><a NAME="E2:58_1_1_2_4_1_1_1"/><span class="kw">then </span>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E2:58_1_1_2"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="ref" href="nat_1.html#T8" onmouseover="rs('nat_1/T8')" onmouseout="rh()">NAT_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:58_1_1_2_4_1_1_1"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E1:58_1_1_2_4"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="txt" href="#E5:58_1_1_2_4"><span class="lab"><font color="Green" title="E45">A15</font></span></a>, <a class="txt" href="#E3:58_1_1_2_4"><span class="lab"><font color="Green" title="E44">A14</font></span></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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_2_4_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:58_1_1_2_4_1_1_2"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E1:58_1_1_2_4"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="txt" href="#E5:58_1_1_2_4"><span class="lab"><font color="Green" title="E45">A15</font></span></a>, <a class="txt" href="#E3:58_1_1_2_4"><span class="lab"><font color="Green" title="E44">A14</font></span></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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:58_1_1_2_4"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</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/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E6:58_1_1_2"/>
( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</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><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_1_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:58_1_1_2_5"/><span class="lab"><font color="Green" title="E43">A16</font></span>: 
<font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> 
 ; <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="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>)</span><br/>

<a NAME="E2:58_1_1_2_5"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1_2"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="ref" href="qc_lang1.html#T16" target="_self" onmouseover="rs('qc_lang1/T16')" onmouseout="rh()">Th16</a></span>;<br/>
<a NAME="E3:58_1_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A17</font></span>: 
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E41">A9</font></span></a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_1_2_5_1"><span class="kw">now </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"> <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_5_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:58_1_1_2_5_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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_5_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_2_5_1_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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>)</span><br/><div class="add"><a NAME="E2:58_1_1_2_5_1_1_1"/><span class="kw">then </span>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E2:58_1_1_2"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="ref" href="nat_1.html#T8" onmouseover="rs('nat_1/T8')" onmouseout="rh()">NAT_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:58_1_1_2_5_1_1_1"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E1:58_1_1_2_5"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E3:58_1_1_2_5"><span class="lab"><font color="Green" title="E44">A17</font></span></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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_5_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_2_5_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:58_1_1_2_5_1_1_2"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E1:58_1_1_2_5"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E3:58_1_1_2_5"><span class="lab"><font color="Green" title="E44">A17</font></span></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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:58_1_1_2_5"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</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/>


</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<a NAME="E2:58_1"/><span class="lab"><font color="Green" title="E39">A18</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c1">F</font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon">F<sub>3</sub></font>() as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon">F<sub>2</sub></font>() ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c1">F</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">S<sub>1</sub></font>[<font color="Maroon" title="c1">F</font>, <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]</span><br/>

<span class="kw">thus </span><a NAME="E2:58_1_2"/>
<font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</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">  for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/>( ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">p</font>) ) &amp; ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  implies <font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive" title="b1">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>) ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">p</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></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"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies ( ( <font color="Maroon" title="c2">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c2">p</font>) ) &amp; ( <font color="Maroon" title="c2">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  implies <font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c2">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c2">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:58_1_2"/><span class="lab"><font color="Green" title="E40">A19</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <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="c2">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c2">p</font>) ) &amp; ( <font color="Maroon" title="c2">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  implies <font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c2">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c2">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>) ) )</span><br/>

<a NAME="E4:58_1_2"/>
1 <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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c2">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#T10" target="_self" onmouseover="rs('qc_lang1/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:58_1_2"/>
( ( <font color="Maroon" title="c2">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c2">p</font>) ) &amp; ( <font color="Maroon" title="c2">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  implies <font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c2">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c2">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1_2"><span class="lab"><font color="Green" title="E40">A19</font></span></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/>


</div><span class="kw">end;</span></div>
<a NAME="E3:58_1"/><span class="lab"><font color="Green" title="E40">A20</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E2:58_1"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="txt" href="#E1:58_1"><span class="lab"><font color="Green" title="E38">A1</font></span></a>);<br/></span>
<a NAME="E4:58_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A21</font></span>: 
 ex <font color="Olive" title="b1">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">G</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span>]
 
;<br/>
<a NAME="E5:58_1"/><span class="lab"><font color="Green" title="E42">A22</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <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="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>() implies  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:58_1_3"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></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">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">x9</font> = <font color="Maroon" title="c1">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon">F<sub>2</sub></font>()<span class="kw"> such that </span><br/><a NAME="E4:58_1_3"/><span class="lab"><font color="Green" title="E43">A23</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">F</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c2">x9</font></span>)</span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1"><span class="lab"><font color="Green" title="E40">A20</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</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">S<sub>4</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c2">x9</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="c2">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; (  for <font color="Olive" title="b1">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon">F<sub>2</sub></font>()  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">g</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c2">x9</font></span>)</span>] holds <br/><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x9</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E5:58_1_3"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x9</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">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon">F<sub>2</sub></font>()  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">g</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c2">x9</font></span>)</span>] holds <br/><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x9</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">G</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon">F<sub>2</sub></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">S<sub>1</sub></font>[<font color="Maroon" title="c4">G</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c2">x9</font></span>)</span>] implies <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x9</font> )</span><br/>

<span class="kw">assume </span><a NAME="E6:58_1_3"/><span class="lab"><font color="Green" title="E44">A24</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">G</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c2">x9</font></span>)</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"> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x9</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>5</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()] <span class="kw">means </span>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> $1</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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c2">x9</font></span>)</span> implies <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 );<br/>
<div><span class="lab"><font color="Green" title="E45">A25</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_3_1"><span class="kw">now </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">  for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">p</font>] ) &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>()] &amp; ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Olive" title="b1">p</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">p</font>] ) &amp; ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Olive" title="b1">p</font>] &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Olive" title="b1">p</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">p</font>] ) &amp; ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Olive" title="b1">p</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">p</font>] ) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">p</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></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="c5">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">p</font>] ) &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>()] &amp; ( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c5">p</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">p</font>] ) &amp; ( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c5">p</font>] &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c5">p</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">p</font>] ) &amp; ( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">p</font>] ) )</span><br/><span class="kw">thus </span><a NAME="E1:58_1_3_1"/>
( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">p</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">S<sub>5</sub></font>[ <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>()] &amp; ( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c5">p</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">p</font>] ) &amp; ( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c5">p</font>] &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c5">p</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">p</font>] ) &amp; ( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">p</font>] ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:58_1_3_1_1"/><span class="lab"><font color="Green" title="E46">A26</font></span>: 
( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c2">x9</font></span>)</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"> <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font></span><br/>

<span class="kw">hence </span><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> = 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c5">p</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_3"><span class="lab"><font color="Green" title="E43">A23</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58_1_3"><span class="lab"><font color="Green" title="E44">A24</font></span></a>, <a class="txt" href="#E1:58_1_3_1_1"><span class="lab"><font color="Green" title="E46">A26</font></span></a></span>
;<br/> <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/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E2:58_1_3_1"/>
<font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>()]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_3"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E6:58_1_3"><span class="lab"><font color="Green" title="E44">A24</font></span></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"> ( ( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c5">p</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">p</font>] ) &amp; ( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c5">p</font>] &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c5">p</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">p</font>] ) &amp; ( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">p</font>] ) )</span><br/><span class="kw">thus </span><a NAME="E3:58_1_3_1"/>
( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c5">p</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">p</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="c5">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c5">p</font>] &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c5">p</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">p</font>] ) &amp; ( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">p</font>] ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_3_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:58_1_3_1_2"/><span class="lab"><font color="Green" title="E46">A27</font></span>: 
<font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> 
 <span class="kw">and </span><br/><a NAME="E2:58_1_3_1_2"/><span class="lab"><font color="Green" title="E47">A28</font></span>: 
<font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c5">p</font>]
 <span class="kw">and </span><br/><a NAME="E3:58_1_3_1_2"/><span class="lab"><font color="Green" title="E48">A29</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c2">x9</font></span>)</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"> <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font></span><br/>

<a NAME="E4:58_1_3_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_3_1_2"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="ref" href="qc_lang1.html#T14" target="_self" onmouseover="rs('qc_lang1/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> = 
<font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_3"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E1:58_1_3_1_2"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="txt" href="#E2:58_1_3_1_2"><span class="lab"><font color="Green" title="E47">A28</font></span></a>, <a class="txt" href="#E3:58_1_3_1_2"><span class="lab"><font color="Green" title="E48">A29</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>
<br/>.= 
<font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58_1_3"><span class="lab"><font color="Green" title="E44">A24</font></span></a>, <a class="txt" href="#E1:58_1_3_1_2"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="txt" href="#E3:58_1_3_1_2"><span class="lab"><font color="Green" title="E48">A29</font></span></a></span>
;<br/> <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/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E4:58_1_3_1"/>
( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c5">p</font>] &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c5">p</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">p</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="c5">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">p</font>] )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_3_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:58_1_3_1_3"/><span class="lab"><font color="Green" title="E46">A30</font></span>: 
<font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> 
 <span class="kw">and </span><br/><a NAME="E2:58_1_3_1_3"/><span class="lab"><font color="Green" title="E47">A31</font></span>: 
( <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c5">p</font>] &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c5">p</font>] )
 <span class="kw">and </span><br/><a NAME="E3:58_1_3_1_3"/><span class="lab"><font color="Green" title="E48">A32</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c2">x9</font></span>)</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"> <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font></span><br/>

<a NAME="E4:58_1_3_1_3"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_3_1_3"><span class="lab"><font color="Green" title="E46">A30</font></span></a>, <a class="ref" href="qc_lang1.html#T15" target="_self" onmouseover="rs('qc_lang1/T15')" onmouseout="rh()">Th15</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> = 
<font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_3"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E1:58_1_3_1_3"><span class="lab"><font color="Green" title="E46">A30</font></span></a>, <a class="txt" href="#E2:58_1_3_1_3"><span class="lab"><font color="Green" title="E47">A31</font></span></a>, <a class="txt" href="#E3:58_1_3_1_3"><span class="lab"><font color="Green" title="E48">A32</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>
<br/>.= 
<font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58_1_3"><span class="lab"><font color="Green" title="E44">A24</font></span></a>, <a class="txt" href="#E1:58_1_3_1_3"><span class="lab"><font color="Green" title="E46">A30</font></span></a>, <a class="txt" href="#E3:58_1_3_1_3"><span class="lab"><font color="Green" title="E48">A32</font></span></a></span>
;<br/> <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/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E5:58_1_3_1"/>
( <font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">p</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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_3_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:58_1_3_1_4"/><span class="lab"><font color="Green" title="E46">A33</font></span>: 
<font color="Maroon" title="c5">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> 
 <span class="kw">and </span><br/><a NAME="E2:58_1_3_1_4"/><span class="lab"><font color="Green" title="E47">A34</font></span>: 
<font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font>]
 <span class="kw">and </span><br/><a NAME="E3:58_1_3_1_4"/><span class="lab"><font color="Green" title="E48">A35</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c2">x9</font></span>)</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"> <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font></span><br/>

<a NAME="E4:58_1_3_1_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c5">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_3_1_4"><span class="lab"><font color="Green" title="E46">A33</font></span></a>, <a class="ref" href="qc_lang1.html#T16" target="_self" onmouseover="rs('qc_lang1/T16')" onmouseout="rh()">Th16</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> = 
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_3"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E1:58_1_3_1_4"><span class="lab"><font color="Green" title="E46">A33</font></span></a>, <a class="txt" href="#E2:58_1_3_1_4"><span class="lab"><font color="Green" title="E47">A34</font></span></a>, <a class="txt" href="#E3:58_1_3_1_4"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>
<br/>.= 
<font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58_1_3"><span class="lab"><font color="Green" title="E44">A24</font></span></a>, <a class="txt" href="#E1:58_1_3_1_4"><span class="lab"><font color="Green" title="E46">A33</font></span></a>, <a class="txt" href="#E3:58_1_3_1_4"><span class="lab"><font color="Green" title="E48">A35</font></span></a></span>
;<br/> <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/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E8:58_1_3"/>
 for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">p</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="qc_lang1.html#S2" target="_self" onmouseover="rs('qc_lang1/S2')" onmouseout="rh()">QC_LANG1:sch 2</a>(<a class="txt" href="#E7:58_1_3"><span class="lab"><font color="Green" title="E45">A25</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E9:58_1_3"/>
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x9</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c1">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E7:58_1"/><span class="lab"><font color="Green" title="E43">A36</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()
 <span class="kw">and </span><br/><a NAME="E8:58_1"/><span class="lab"><font color="Green" title="E44">A37</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="classes1.html#S1" onmouseover="rs('classes1/S1')" onmouseout="rh()">CLASSES1:sch 1</a>(<a class="txt" href="#E5:58_1"><span class="lab"><font color="Green" title="E42">A22</font></span></a>);<br/></span>
<a NAME="E9:58_1"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>()
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <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"> (  not <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c1">F</font> or <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() )</span><br/>

<span class="kw">assume </span><a NAME="E1:58_1_4"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c1">F</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="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>()</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:58_1_4"/><span class="lab"><font color="Green" title="E45">A38</font></span>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1"><span class="lab"><font color="Green" title="E43">A36</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()<span class="kw"> such that </span><br/><a NAME="E5:58_1_4"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">and </span><br/><a NAME="E6:58_1_4"/><span class="lab"><font color="Green" title="E46">A39</font></span>: 
 for <font color="Olive" title="b1">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon">F<sub>2</sub></font>()  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">g</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c4">p</font></span>)</span>] holds <br/><font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1"><span class="lab"><font color="Green" title="E44">A37</font></span></a>, <a class="txt" href="#E3:58_1_4"><span class="lab"><font color="Green" title="E45">A38</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon">F<sub>2</sub></font>()<span class="kw"> such that </span><br/><a NAME="E8:58_1_4"/><span class="lab"><font color="Green" title="E47">A40</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">G</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c4">p</font></span>)</span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1"><span class="lab"><font color="Green" title="E40">A20</font></span></a></span>;<br/>
<a NAME="E9:58_1_4"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58_1_4"><span class="lab"><font color="Green" title="E46">A39</font></span></a>, <a class="txt" href="#E8:58_1_4"><span class="lab"><font color="Green" title="E47">A40</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:58_1_4"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">F</font> = <font color="Maroon" title="c1">F</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon">F<sub>2</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1"><span class="lab"><font color="Green" title="E43">A36</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" onmouseover="rs('relset_1/T4')" onmouseout="rh()">RELSET_1:4</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c2">F</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="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() &amp; (  for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">p</font>) ) &amp; ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive" title="b1">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>) ) ) ) )</span><br/>

<a NAME="E11:58_1"/>
<font color="Maroon">S<sub>4</sub></font>[ <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1"><span class="lab"><font color="Green" title="E44">A37</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:58_1"/>
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1"><span class="lab"><font color="Green" title="E41">A21</font></span></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">  for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">p</font>) ) &amp; ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive" title="b1">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>) ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">p</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></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">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c3">p</font>) ) &amp; ( <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>) ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">p1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()<span class="kw"> such that </span><br/><a NAME="E14:58_1"/><span class="lab"><font color="Green" title="E45">A41</font></span>: 
<font color="Maroon" title="c4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font>
 <span class="kw">and </span><br/><a NAME="E15:58_1"/><span class="lab"><font color="Green" title="E46">A42</font></span>: 
 for <font color="Olive" title="b1">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon">F<sub>2</sub></font>()  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">g</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c4">p1</font></span>)</span>] holds <br/><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1"><span class="lab"><font color="Green" title="E44">A37</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon">F<sub>2</sub></font>()<span class="kw"> such that </span><br/><a NAME="E17:58_1"/><span class="lab"><font color="Green" title="E47">A43</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">G</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c4">p1</font></span>)</span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1"><span class="lab"><font color="Green" title="E40">A20</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">p9</font> =  <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c3">p</font>;<br/>
<a NAME="E18:58_1"/><span class="lab"><font color="Green" title="E48">A44</font></span>: 
 ex <font color="Olive" title="b1">p1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c3">p</font> &amp; (  for <font color="Olive" title="b2">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon">F<sub>2</sub></font>()  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">g</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Olive" title="b1">p1</font></span>)</span>] holds <br/><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p1</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1"><span class="lab"><font color="Green" title="E44">A37</font></span></a></span>;<br/>
<a NAME="E19:58_1"/><span class="lab"><font color="Green" title="E49">A45</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:58_1"><span class="lab"><font color="Green" title="E45">A41</font></span></a>, <a class="txt" href="#E15:58_1"><span class="lab"><font color="Green" title="E46">A42</font></span></a>, <a class="txt" href="#E17:58_1"><span class="lab"><font color="Green" title="E47">A43</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:58_1"/>
( <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c3">p</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:58_1"><span class="lab"><font color="Green" title="E45">A41</font></span></a>, <a class="txt" href="#E17:58_1"><span class="lab"><font color="Green" title="E47">A43</font></span></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"> ( ( <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>) ) )</span><br/>

<a NAME="E21:58_1"/><span class="lab"><font color="Green" title="E50">A46</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">G</font>,<font color="Olive" title="b1">k</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <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="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">G</font>,<font color="Maroon" title="c7">k</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:58_1_5"/><span class="lab"><font color="Green" title="E51">A47</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</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"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">G</font>,<font color="Maroon" title="c7">k</font>]</span><br/>

<span class="kw">thus </span><a NAME="E2:58_1_5"/>
<font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:58_1"><span class="lab"><font color="Green" title="E47">A43</font></span></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">  for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> holds <br/>( ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">p</font>) ) &amp; ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  implies <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Olive" title="b1">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive" title="b1">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>) ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">p9</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></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"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c8">p9</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> implies ( ( <font color="Maroon" title="c8">p9</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c8">p9</font>) ) &amp; ( <font color="Maroon" title="c8">p9</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  implies <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c8">p9</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c8">p9</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c8">p9</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c8">p9</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c8">p9</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c8">p9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c8">p9</font></span>)</span></span>)</span>) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:58_1_5"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c8">p9</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</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="c8">p9</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c8">p9</font>) ) &amp; ( <font color="Maroon" title="c8">p9</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  implies <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c8">p9</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c8">p9</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c8">p9</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c8">p9</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c8">p9</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c8">p9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c8">p9</font></span>)</span></span>)</span>) ) )</span><br/>

<a NAME="E4:58_1_5"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c8">p9</font></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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_5"><span class="lab"><font color="Green" title="E51">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:58_1_5"/>
( ( <font color="Maroon" title="c8">p9</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c8">p9</font>) ) &amp; ( <font color="Maroon" title="c8">p9</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  implies <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c8">p9</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c8">p9</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c8">p9</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c8">p9</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c8">p9</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c8">p9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c8">p9</font></span>)</span></span>)</span>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:58_1"><span class="lab"><font color="Green" title="E45">A41</font></span></a>, <a class="txt" href="#E17:58_1"><span class="lab"><font color="Green" title="E47">A43</font></span></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/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E22:58_1"/>
( <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</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"> ( ( <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>) ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c7">p9</font> =  <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c3">p</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">k</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:58_1_6"/><span class="lab"><font color="Green" title="E51">A48</font></span>: 
 ex <font color="Olive" title="b1">p1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c3">p</font> &amp; (  for <font color="Olive" title="b2">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon">F<sub>2</sub></font>()  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">g</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Olive" title="b1">p1</font></span>)</span>] holds <br/><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p1</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1"><span class="lab"><font color="Green" title="E44">A37</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:58_1_6"/><span class="lab"><font color="Green" title="E52">A49</font></span>: 
<font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> 
 ; <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="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>)</span><br/>

<a NAME="E3:58_1_6"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#T14" target="_self" onmouseover="rs('qc_lang1/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<a NAME="E4:58_1_6"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">G</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:58_1"><span class="lab"><font color="Green" title="E50">A46</font></span></a></span>;<br/>
<a NAME="E5:58_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_6"><span class="lab"><font color="Green" title="E51">A48</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:58_1_6"/>
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:58_1"><span class="lab"><font color="Green" title="E45">A41</font></span></a>, <a class="txt" href="#E17:58_1"><span class="lab"><font color="Green" title="E47">A43</font></span></a>, <a class="txt" href="#E19:58_1"><span class="lab"><font color="Green" title="E49">A45</font></span></a>, <a class="txt" href="#E2:58_1_6"><span class="lab"><font color="Green" title="E52">A49</font></span></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/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E23:58_1"/>
( <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</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"> ( <font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c7">p99</font> =  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c3">p</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">p9</font> =  <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c3">p</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">k9</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">k99</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:58_1_7"/><span class="lab"><font color="Green" title="E51">A50</font></span>: 
 ex <font color="Olive" title="b1">p2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive" title="b1">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c3">p</font> &amp; (  for <font color="Olive" title="b2">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon">F<sub>2</sub></font>()  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">g</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Olive" title="b1">p2</font></span>)</span>] holds <br/><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p2</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1"><span class="lab"><font color="Green" title="E44">A37</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:58_1_7"/><span class="lab"><font color="Green" title="E52">A51</font></span>: 
<font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> 
 ; <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="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>)</span><br/>

<a NAME="E3:58_1_7"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#T15" target="_self" onmouseover="rs('qc_lang1/T15')" onmouseout="rh()">Th15</a></span>;<br/>
<a NAME="E4:58_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A52</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">G</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:58_1"><span class="lab"><font color="Green" title="E50">A46</font></span></a></span>;<br/>
<a NAME="E5:58_1_7"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_7"><span class="lab"><font color="Green" title="E52">A51</font></span></a>, <a class="ref" href="qc_lang1.html#T15" target="_self" onmouseover="rs('qc_lang1/T15')" onmouseout="rh()">Th15</a></span>;<br/>
<a NAME="E6:58_1_7"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">G</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:58_1"><span class="lab"><font color="Green" title="E50">A46</font></span></a></span>;<br/>
<a NAME="E7:58_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A53</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_7"><span class="lab"><font color="Green" title="E51">A50</font></span></a></span>;<br/>
<a NAME="E8:58_1_7"/>
 ex <font color="Olive" title="b1">p1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c3">p</font> &amp; (  for <font color="Olive" title="b2">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon">F<sub>2</sub></font>()  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">g</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Olive" title="b1">p1</font></span>)</span>] holds <br/><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p1</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1"><span class="lab"><font color="Green" title="E44">A37</font></span></a></span>;<br/>
<a NAME="E9:58_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_7"><span class="lab"><font color="Green" title="E53">A52</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:58_1_7"/>
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:58_1"><span class="lab"><font color="Green" title="E45">A41</font></span></a>, <a class="txt" href="#E17:58_1"><span class="lab"><font color="Green" title="E47">A43</font></span></a>, <a class="txt" href="#E19:58_1"><span class="lab"><font color="Green" title="E49">A45</font></span></a>, <a class="txt" href="#E2:58_1_7"><span class="lab"><font color="Green" title="E52">A51</font></span></a>, <a class="txt" href="#E7:58_1_7"><span class="lab"><font color="Green" title="E54">A53</font></span></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/>


</div><span class="kw">end;</span></div>

<span class="kw">set </span><font color="Maroon" title="c7">k</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E24:58_1"/><span class="lab"><font color="Green" title="E51">A54</font></span>: 
<font color="Maroon" title="c3">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> 
 ; <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="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>)</span><br/>

<a NAME="E25:58_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#T16" target="_self" onmouseover="rs('qc_lang1/T16')" onmouseout="rh()">Th16</a></span>;<br/>
<a NAME="E26:58_1"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">G</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:58_1"><span class="lab"><font color="Green" title="E50">A46</font></span></a></span>;<br/>
<a NAME="E27:58_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c3">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:58_1"><span class="lab"><font color="Green" title="E48">A44</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E28:58_1"/>
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:58_1"><span class="lab"><font color="Green" title="E45">A41</font></span></a>, <a class="txt" href="#E17:58_1"><span class="lab"><font color="Green" title="E47">A43</font></span></a>, <a class="txt" href="#E19:58_1"><span class="lab"><font color="Green" title="E49">A45</font></span></a>, <a class="txt" href="#E24:58_1"><span class="lab"><font color="Green" title="E51">A54</font></span></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/>


</div>
