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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">F</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</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">  for <font color="Olive" title="b1">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>   st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a>  holds <br/> <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a>  <a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio2.html#K2" title="LTLAXIO2:func.2">con</a> <span class="p3">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="ltlaxio2.html#K2" title="LTLAXIO2:func.2">con</a> <span class="p4">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</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"> (  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a>  implies  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a>  <a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio2.html#K2" title="LTLAXIO2:func.2">con</a> <span class="p3">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="ltlaxio2.html#K2" title="LTLAXIO2:func.2">con</a> <span class="p4">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span>)</span></span>)</span> )</span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">F</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>  st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> for <font color="Olive" title="b2">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>   st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a>  holds <br/> <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">f</font>);<br/>
<span class="kw">assume </span><a NAME="E1:33"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.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">  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a>  <a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio2.html#K2" title="LTLAXIO2:func.2">con</a> <span class="p3">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="ltlaxio2.html#K2" title="LTLAXIO2:func.2">con</a> <span class="p4">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span>)</span></span>)</span></span><br/>

<a NAME="E2:33"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c3">F</font></span>)</span>
 
;<br/>
<div><span class="lab"><font color="Green" title="E20">A3</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_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">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">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">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:33_1"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">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">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><span class="kw">thus </span><a NAME="E2:33_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</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="33_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">G</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</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"> (  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies  for <font color="Olive" title="b1">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>   st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a>  holds <br/> <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">f</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_1_1"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</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">  for <font color="Olive" title="b1">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>   st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a>  holds <br/> <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">f</font>)</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">H</font> = <font color="Maroon" title="c6">G</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">g</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</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"> (  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a>  implies  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">g</font>) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c9">A</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <span class="kw"> such that </span><br/><a NAME="E4:33_1_1"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
<font color="Maroon" title="c9">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font>
 <span class="kw">and </span><br/><a NAME="E5:33_1_1"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
 <a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#T15" target="_self" onmouseover="rs('ltlaxio4/T15')" onmouseout="rh()">Th15</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">Fp</font> = <span class="p1">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">ff</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E7:33_1_1"/><span class="lab"><font color="Green" title="E25">A8</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:33_1_1"/>
<font color="Maroon" title="c11">ff</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T58" onmouseover="rs('finseq_4/T58')" onmouseout="rh()">FINSEQ_4:58</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">ff</font> = <font color="Maroon" title="c11">ff</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio3.html#K1" title="LTLAXIO3:func.1">**</a></span>)</span>,<span class="p2">(<span class="default"><a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio3.html#K1" title="LTLAXIO3:func.1">**</a></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33_1_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>  = <span class="p1"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">]</a></span> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a> ;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>4</sub></font>(  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>  = <span class="p1"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">]</a></span> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">ff1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <span class="kw"> such that </span><br/><a NAME="E11:33_1_1"/><span class="lab"><font color="Green" title="E26">A9</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">ff1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">ff</font> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">ff1</font> holds <br/><font color="Maroon" title="c13">ff1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">i</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_4.html#S2" onmouseover="rs('finseq_4/S2')" onmouseout="rh()">FINSEQ_4:sch 2</a>();<br/></span>
<span class="kw">consider </span><font color="Maroon" title="c14">ff2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <span class="kw"> such that </span><br/><a NAME="E13:33_1_1"/><span class="lab"><font color="Green" title="E27">A10</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">ff2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">ff</font> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">ff2</font> holds <br/><font color="Maroon" title="c14">ff2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">i</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_4.html#S2" onmouseover="rs('finseq_4/S2')" onmouseout="rh()">FINSEQ_4:sch 2</a>();<br/></span>
<span class="kw">set </span><font color="Maroon" title="c15">fl</font> = <font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>;<br/>
<div><span class="lab"><font color="Green" title="E28">A11</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_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">  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st ( <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">ff1</font> or <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">ff2</font> ) holds <br/>(  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">i</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="c16">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">ff1</font> or <font color="Maroon" title="c16">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">ff2</font> ) implies (  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> ) )</span><br/><span class="kw">set </span><font color="Maroon" title="c17">Q</font> = <font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font>;<br/><span class="kw">assume </span><a NAME="E1:33_1_1_1"/>
( <font color="Maroon" title="c16">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">ff1</font> or <font color="Maroon" title="c16">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">ff2</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="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> )</span><br/><a NAME="E2:33_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">ff</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:33_1_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E13:33_1_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="ref" href="finseq_3.html#T29" onmouseover="rs('finseq_3/T29')" onmouseout="rh()">FINSEQ_3:29</a></span>;<br/><a NAME="E3:33_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a>, <a class="txt" href="#E7:33_1_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a></span>;<br/><a NAME="E4:33_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">R</font> being   <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</a> st <br/>( <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font> &amp;  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">R</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">R</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> )
 ;<br/><span class="kw">hence </span><a NAME="E5:33_1_1_1"/>
(  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></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>
<a NAME="E15:33_1_1"/>
 <a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" onmouseover="rs('xboole_1/T79')" onmouseout="rh()">XBOOLE_1:79</a>, <a class="txt" href="#E5:33_1_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a></span>;<br/>
<a NAME="E16:33_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A12</font></span>: 
 <a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T38" onmouseover="rs('finseq_1/T38')" onmouseout="rh()">FINSEQ_1:38</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E30">A13</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_1_2"><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">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">ff1</font> holds <br/>(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">i</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="c16">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">ff1</font> implies (  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )</span><br/><span class="kw">set </span><font color="Maroon" title="c17">Q</font> = <font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font>;<br/><span class="kw">assume </span><a NAME="E1:33_1_1_2"/>
<font color="Maroon" title="c16">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">ff1</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="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><a NAME="E2:33_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A14</font></span>: 
 <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:33_1_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E14:33_1_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:33_1_1_2"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T63" onmouseover="rs('xboole_1/T63')" onmouseout="rh()">XBOOLE_1:63</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">  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">thus </span><a NAME="E4:33_1_1_2"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a>, <a class="txt" href="#E2:33_1_1_2"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="ref" href="xboole_1.html#T63" onmouseover="rs('xboole_1/T63')" onmouseout="rh()">XBOOLE_1:63</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="lab"><font color="Green" title="E31">A15</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_1_3"><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">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">ff2</font> holds <br/>(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">i</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="c16">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">ff2</font> implies (  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )</span><br/><span class="kw">set </span><font color="Maroon" title="c17">Q</font> = <font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font>;<br/><span class="kw">assume </span><a NAME="E1:33_1_1_3"/>
<font color="Maroon" title="c16">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">ff2</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="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><a NAME="E2:33_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A16</font></span>: 
 <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:33_1_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E14:33_1_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:33_1_1_3"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T63" onmouseover="rs('xboole_1/T63')" onmouseout="rh()">XBOOLE_1:63</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">  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">thus </span><a NAME="E4:33_1_1_3"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a>, <a class="txt" href="#E2:33_1_1_3"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="xboole_1.html#T63" onmouseover="rs('xboole_1/T63')" onmouseout="rh()">XBOOLE_1:63</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="E19:33_1_1"/><span class="lab"><font color="Green" title="E32">A17</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">x</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="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span> or <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_1_1_4"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</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="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a> </span><br/>

<a NAME="E2:33_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A18</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c13">ff1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c14">ff2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T31" onmouseover="rs('finseq_1/T31')" onmouseout="rh()">FINSEQ_1:31</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_4_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="E1:33_1_1_4_1"/>
( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c13">ff1</font> or <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c14">ff2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_1_1_4"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:33_1_1_4_1_1"/><span class="lab"><font color="Green" title="E34">A19</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c13">ff1</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="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a> </span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font>;<br/><span class="kw">set </span><font color="Maroon" title="c18">Q</font> = <font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c19">P1</font> = <span class="p1"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">]</a></span>;<br/><a NAME="E2:33_1_1_4_1_1"/><span class="lab"><font color="Green" title="E35">A20</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">ff1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1_1_4_1_1"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="finseq_4.html#T20" onmouseover="rs('finseq_4/T20')" onmouseout="rh()">FINSEQ_4:20</a></span>;<br/><a NAME="E3:33_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A21</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:33_1_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a></span>;<br/><a NAME="E4:33_1_1_4_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:33_1_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="txt" href="#E2:33_1_1_4_1_1"><span class="lab"><font color="Green" title="E35">A20</font></span></a></span>;<br/><a NAME="E5:33_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A22</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">]</a></span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T38" onmouseover="rs('finseq_1/T38')" onmouseout="rh()">FINSEQ_1:38</a></span>;<br/><span class="lab"><font color="Green" title="E38">A23</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">]</a></span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> = 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio3.html#D3" onmouseover="rs('ltlaxio3/D3')" onmouseout="rh()">LTLAXIO3:def 3</a>, <a class="txt" href="#E3:33_1_1_4_1_1"><span class="lab"><font color="Green" title="E36">A21</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T31" onmouseover="rs('finseq_1/T31')" onmouseout="rh()">FINSEQ_1:31</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T38" onmouseover="rs('finseq_1/T38')" onmouseout="rh()">FINSEQ_1:38</a></span>
;<br/><a NAME="E7:33_1_1_4_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:33_1_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E2:33_1_1_4_1_1"><span class="lab"><font color="Green" title="E35">A20</font></span></a></span>;<br/><a NAME="E8:33_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A24</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">]</a></span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">]</a></span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T70" onmouseover="rs('xboole_1/T70')" onmouseout="rh()">XBOOLE_1:70</a>, <a class="txt" href="#E5:33_1_1_4_1_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="txt" href="#E6:33_1_1_4_1_1"><span class="lab"><font color="Green" title="E38">A23</font></span></a></span>;<br/> <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p1"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">]</a></span> = 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T4" onmouseover="rs('xboole_1/T4')" onmouseout="rh()">XBOOLE_1:4</a>, <a class="txt" href="#E6:33_1_1_4_1_1"><span class="lab"><font color="Green" title="E38">A23</font></span></a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:33_1_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E2:33_1_1_4_1_1"><span class="lab"><font color="Green" title="E35">A20</font></span></a></span>
<br/>.= 
 <a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c6">G</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T45" onmouseover="rs('xboole_1/T45')" onmouseout="rh()">XBOOLE_1:45</a>, <a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a>, <a class="txt" href="#E4:33_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E5:33_1_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a></span>
;<br/><a NAME="E10:33_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A25</font></span>: 
<span class="p1"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <font color="Maroon" title="c6">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33_1_1_4_1_1"><span class="lab"><font color="Green" title="E39">A24</font></span></a></span>;<br/><font color="Maroon" title="c16">x</font> = 
<font color="Maroon" title="c13">ff1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T38" onmouseover="rs('finseq_5/T38')" onmouseout="rh()">FINSEQ_5:38</a>, <a class="txt" href="#E1:33_1_1_4_1_1"><span class="lab"><font color="Green" title="E34">A19</font></span></a></span>
<br/>.= 
<span class="p1"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">]</a></span> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:33_1_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E1:33_1_1_4_1_1"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="finseq_4.html#T20" onmouseover="rs('finseq_4/T20')" onmouseout="rh()">FINSEQ_4:20</a></span>
;<br/><span class="kw">hence </span><a NAME="E12:33_1_1_4_1_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:33_1_1_4_1_1"><span class="lab"><font color="Green" title="E40">A25</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="33_1_1_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:33_1_1_4_1_2"/><span class="lab"><font color="Green" title="E34">A26</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c14">ff2</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="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a> </span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font>;<br/><span class="kw">set </span><font color="Maroon" title="c18">Q</font> = <font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c19">P1</font> = <span class="p1"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">]</a></span>;<br/><a NAME="E2:33_1_1_4_1_2"/><span class="lab"><font color="Green" title="E35">A27</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">ff2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1_1_4_1_2"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="ref" href="finseq_4.html#T20" onmouseover="rs('finseq_4/T20')" onmouseout="rh()">FINSEQ_4:20</a></span>;<br/><a NAME="E3:33_1_1_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A28</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:33_1_1"><span class="lab"><font color="Green" title="E31">A15</font></span></a></span>;<br/><a NAME="E4:33_1_1_4_1_2"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:33_1_1"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="txt" href="#E2:33_1_1_4_1_2"><span class="lab"><font color="Green" title="E35">A27</font></span></a></span>;<br/><a NAME="E5:33_1_1_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A29</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">]</a></span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T38" onmouseover="rs('finseq_1/T38')" onmouseout="rh()">FINSEQ_1:38</a></span>;<br/><span class="lab"><font color="Green" title="E38">A30</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">]</a></span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> = 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio3.html#D3" onmouseover="rs('ltlaxio3/D3')" onmouseout="rh()">LTLAXIO3:def 3</a>, <a class="txt" href="#E3:33_1_1_4_1_2"><span class="lab"><font color="Green" title="E36">A28</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T31" onmouseover="rs('finseq_1/T31')" onmouseout="rh()">FINSEQ_1:31</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T38" onmouseover="rs('finseq_1/T38')" onmouseout="rh()">FINSEQ_1:38</a></span>
;<br/><a NAME="E7:33_1_1_4_1_2"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:33_1_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E2:33_1_1_4_1_2"><span class="lab"><font color="Green" title="E35">A27</font></span></a></span>;<br/><a NAME="E8:33_1_1_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A31</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">]</a></span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">]</a></span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T70" onmouseover="rs('xboole_1/T70')" onmouseout="rh()">XBOOLE_1:70</a>, <a class="txt" href="#E5:33_1_1_4_1_2"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E6:33_1_1_4_1_2"><span class="lab"><font color="Green" title="E38">A30</font></span></a></span>;<br/> <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p1"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">]</a></span> = 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T4" onmouseover="rs('xboole_1/T4')" onmouseout="rh()">XBOOLE_1:4</a>, <a class="txt" href="#E6:33_1_1_4_1_2"><span class="lab"><font color="Green" title="E38">A30</font></span></a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:33_1_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E2:33_1_1_4_1_2"><span class="lab"><font color="Green" title="E35">A27</font></span></a></span>
<br/>.= 
 <a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c6">G</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T45" onmouseover="rs('xboole_1/T45')" onmouseout="rh()">XBOOLE_1:45</a>, <a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a>, <a class="txt" href="#E4:33_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E5:33_1_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a></span>
;<br/><a NAME="E10:33_1_1_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A32</font></span>: 
<span class="p1"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <font color="Maroon" title="c6">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33_1_1_4_1_2"><span class="lab"><font color="Green" title="E39">A31</font></span></a></span>;<br/><font color="Maroon" title="c16">x</font> = 
<font color="Maroon" title="c14">ff2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T38" onmouseover="rs('finseq_5/T38')" onmouseout="rh()">FINSEQ_5:38</a>, <a class="txt" href="#E1:33_1_1_4_1_2"><span class="lab"><font color="Green" title="E34">A26</font></span></a></span>
<br/>.= 
<span class="p1"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">]</a></span> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:33_1_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E1:33_1_1_4_1_2"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="ref" href="finseq_4.html#T20" onmouseover="rs('finseq_4/T20')" onmouseout="rh()">FINSEQ_4:20</a></span>
;<br/><span class="kw">hence </span><a NAME="E12:33_1_1_4_1_2"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:33_1_1_4_1_2"><span class="lab"><font color="Green" title="E40">A32</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">assume </span><a NAME="E20:33_1_1"/><span class="lab"><font color="Green" title="E33">A33</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.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">  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">g</font>)</span><br/>

<a NAME="E21:33_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>) <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">g</font>) is  <a href="ltlaxio2.html#NV1" title="LTLAXIO2:NV.1">ctaut</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">h</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>,<a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ltlaxio1.html#D16" onmouseover="rs('ltlaxio1/D16')" onmouseout="rh()">LTLAXIO1:def 16</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"> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>) <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">g</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<span class="kw">set </span><font color="Maroon" title="c17">v</font> =  <a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c16">h</font>;<br/>
<a NAME="E1:33_1_1_5"/><span class="lab"><font color="Green" title="E34">A34</font></span>: 
( <span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboolean.html#D3" onmouseover="rs('xboolean/D3')" onmouseout="rh()">XBOOLEAN:def 3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E35">A35</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_1_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"> ( <span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  not <span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:33_1_1_5_1"/><span class="lab"><font color="Green" title="E36">A36</font></span>: 
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E2:33_1_1_5_1"/><span class="lab"><font color="Green" title="E37">A37</font></span>: 
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_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="E1:33_1_1_5_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_5_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:33_1_1_5_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> contradiction</span><br/><div class="add"><a NAME="E2:33_1_1_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="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#D4" onmouseover="rs('ltlaxio2/D4')" onmouseout="rh()">LTLAXIO2:def 4</a></span>;<br/><a NAME="E3:33_1_1_5_1_1_1"/><span class="kw">then </span>
 <a href="ltlaxio2.html#K2" title="LTLAXIO2:func.2">con</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="ltlaxio1.html#K3" title="LTLAXIO1:func.3">TVERUM</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#D2" onmouseover="rs('ltlaxio2/D2')" onmouseout="rh()">LTLAXIO2:def 2</a></span>;<br/><span class="kw">then </span><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>) = 
 <a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="ltlaxio1.html#K3" title="LTLAXIO1:func.3">TVERUM</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T39" onmouseover="rs('finseq_1/T39')" onmouseout="rh()">FINSEQ_1:39</a></span>
<br/>.= 
 <a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <a href="ltlaxio1.html#K3" title="LTLAXIO1:func.3">TVERUM</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T16" onmouseover="rs('finseq_4/T16')" onmouseout="rh()">FINSEQ_4:16</a></span>
;<br/><span class="kw">then </span><span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>) = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="ltlaxio1.html#K3" title="LTLAXIO1:func.3">TVERUM</a></span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="ltlaxio1.html#NK2" title="LTLAXIO1:NK.2">TFALSUM</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D15" onmouseover="rs('ltlaxio1/D15')" onmouseout="rh()">LTLAXIO1:def 15</a></span>
<br/>.= 
<a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="ltlaxio1.html#NK2" title="LTLAXIO1:NK.2">TFALSUM</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T4" onmouseover="rs('ltlaxio2/T4')" onmouseout="rh()">LTLAXIO2:4</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:33_1_1_5_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1_1_5_1"><span class="lab"><font color="Green" title="E36">A36</font></span></a>, <a class="ref" href="ltlaxio1.html#D15" onmouseover="rs('ltlaxio1/D15')" onmouseout="rh()">LTLAXIO1:def 15</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="33_1_1_5_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:33_1_1_5_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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"> contradiction</span><br/><div class="add"><a NAME="E2:33_1_1_5_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span>)</span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="ltlaxio1.html#NK2" title="LTLAXIO1:NK.2">TFALSUM</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1_1_5_1"><span class="lab"><font color="Green" title="E36">A36</font></span></a>, <a class="ref" href="ltlaxio1.html#D15" onmouseover="rs('ltlaxio1/D15')" onmouseout="rh()">LTLAXIO1:def 15</a></span>;<br/><a NAME="E3:33_1_1_5_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span>)</span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D15" onmouseover="rs('ltlaxio1/D15')" onmouseout="rh()">LTLAXIO1:def 15</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:33_1_1_5_1_1_2"/><span class="lab"><font color="Green" title="E38">A38</font></span>: 
<font color="Maroon" title="c18">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:33_1_1_5_1_1_2"/><span class="lab"><font color="Green" title="E39">A39</font></span>: 
 not <span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T19" onmouseover="rs('ltlaxio2/T19')" onmouseout="rh()">LTLAXIO2:19</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">i1</font> = <font color="Maroon" title="c18">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><a NAME="E8:33_1_1_5_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#D4" onmouseover="rs('ltlaxio2/D4')" onmouseout="rh()">LTLAXIO2:def 4</a></span>;<br/><a NAME="E9:33_1_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A40</font></span>: 
<font color="Maroon" title="c19">i1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" onmouseover="rs('finseq_3/T29')" onmouseout="rh()">FINSEQ_3:29</a>, <a class="txt" href="#E5:33_1_1_5_1_1_2"><span class="lab"><font color="Green" title="E38">A38</font></span></a></span>;<br/><a NAME="E10:33_1_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A41</font></span>: 
 not <span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T8" onmouseover="rs('ltlaxio2/T8')" onmouseout="rh()">LTLAXIO2:8</a>, <a class="txt" href="#E6:33_1_1_5_1_1_2"><span class="lab"><font color="Green" title="E39">A39</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c20">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c8">g</font>;<br/><a NAME="E11:33_1_1_5_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c8">g</font>)</span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="ltlaxio1.html#NK2" title="LTLAXIO1:NK.2">TFALSUM</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_1_1_5_1"><span class="lab"><font color="Green" title="E37">A37</font></span></a>, <a class="ref" href="ltlaxio1.html#D15" onmouseover="rs('ltlaxio1/D15')" onmouseout="rh()">LTLAXIO1:def 15</a></span>;<br/><a NAME="E12:33_1_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A42</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c8">g</font>)</span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D15" onmouseover="rs('ltlaxio1/D15')" onmouseout="rh()">LTLAXIO1:def 15</a></span>;<br/><a NAME="E13:33_1_1_5_1_1_2"/><span class="lab"><font color="Green" title="E43">A43</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a>, <a class="txt" href="#E9:33_1_1_5_1_1_2"><span class="lab"><font color="Green" title="E40">A40</font></span></a></span>;<br/><a NAME="E14:33_1_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A44</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c8">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:33_1_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E20:33_1_1"><span class="lab"><font color="Green" title="E33">A33</font></span></a>, <a class="ref" href="finseq_4.html#T20" onmouseover="rs('finseq_4/T20')" onmouseout="rh()">FINSEQ_4:20</a></span>;<br/><a NAME="E15:33_1_1_5_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c8">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E16:33_1_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A45</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c8">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c8">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#D4" onmouseover="rs('ltlaxio2/D4')" onmouseout="rh()">LTLAXIO2:def 4</a></span>;<br/><a NAME="E17:33_1_1_5_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c8">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:33_1_1_5_1_1_2"><span class="lab"><font color="Green" title="E44">A44</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E18:33_1_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A46</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c8">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c8">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:33_1_1_5_1_1_2"><span class="lab"><font color="Green" title="E45">A45</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c8">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c8">g</font></span>)</span> = 
 <a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c8">g</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T8" onmouseover="rs('ltlaxio2/T8')" onmouseout="rh()">LTLAXIO2:8</a>, <a class="txt" href="#E14:33_1_1_5_1_1_2"><span class="lab"><font color="Green" title="E44">A44</font></span></a></span>
<br/>.= 
 <a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T38" onmouseover="rs('finseq_5/T38')" onmouseout="rh()">FINSEQ_5:38</a>, <a class="txt" href="#E13:33_1_1_5_1_1_2"><span class="lab"><font color="Green" title="E43">A43</font></span></a>, <a class="txt" href="#E19:33_1_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E20:33_1_1"><span class="lab"><font color="Green" title="E33">A33</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E20:33_1_1_5_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:33_1_1_5_1_1_2"><span class="lab"><font color="Green" title="E42">A42</font></span></a>, <a class="txt" href="#E18:33_1_1_5_1_1_2"><span class="lab"><font color="Green" title="E46">A46</font></span></a>, <a class="ref" href="ltlaxio2.html#T19" onmouseover="rs('ltlaxio2/T19')" onmouseout="rh()">LTLAXIO2:19</a>, <a class="txt" href="#E10:33_1_1_5_1_1_2"><span class="lab"><font color="Green" title="E41">A41</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">thus </span><span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>) <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">g</font>)</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>)</span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">g</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D15" onmouseover="rs('ltlaxio1/D15')" onmouseout="rh()">LTLAXIO1:def 15</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_1_1_5"><span class="lab"><font color="Green" title="E35">A35</font></span></a>, <a class="txt" href="#E1:33_1_1_5"><span class="lab"><font color="Green" title="E34">A34</font></span></a>, <a class="ref" href="xboolean.html#D3" onmouseover="rs('xboolean/D3')" onmouseout="rh()">XBOOLEAN:def 3</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="E22:33_1_1"/><span class="kw">then </span>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>) <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">g</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio1.html#K13" title="LTLAXIO1:func.13">LTL_axioms</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D17" onmouseover="rs('ltlaxio1/D17')" onmouseout="rh()">LTLAXIO1:def 17</a></span>;<br/>
<a NAME="E23:33_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A47</font></span>: 
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>) <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">g</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#T42" onmouseover="rs('ltlaxio1/T42')" onmouseout="rh()">LTLAXIO1:42</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>5</sub></font>(  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">fk</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <span class="kw"> such that </span><br/><a NAME="E25:33_1_1"/><span class="lab"><font color="Green" title="E35">A48</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">ff</font> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">fk</font> holds <br/><font color="Maroon" title="c16">fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">i</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_4.html#S2" onmouseover="rs('finseq_4/S2')" onmouseout="rh()">FINSEQ_4:sch 2</a>();<br/></span>
<div><span class="lab"><font color="Green" title="E36">A49</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_1_6"><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">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>,<a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a>  st <span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">fk</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/><span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>,<a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</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"> ( <span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">fk</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">set </span><font color="Maroon" title="c18">v</font> =  <a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font>;<br/><span class="kw">assume </span><a NAME="E1:33_1_1_6"/>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">fk</font>) <a href="hidden.html#R1" title="HIDDEN:pred.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"> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E2:33_1_1_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c16">fk</font>)</span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="ltlaxio1.html#NK2" title="LTLAXIO1:NK.2">TFALSUM</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D15" onmouseover="rs('ltlaxio1/D15')" onmouseout="rh()">LTLAXIO1:def 15</a></span>;<br/><a NAME="E3:33_1_1_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c16">fk</font>)</span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D15" onmouseover="rs('ltlaxio1/D15')" onmouseout="rh()">LTLAXIO1:def 15</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:33_1_1_6"/><span class="lab"><font color="Green" title="E37">A50</font></span>: 
<font color="Maroon" title="c19">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c16">fk</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:33_1_1_6"/><span class="lab"><font color="Green" title="E38">A51</font></span>: 
 not <span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c16">fk</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T19" onmouseover="rs('ltlaxio2/T19')" onmouseout="rh()">LTLAXIO2:19</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">i1</font> = <font color="Maroon" title="c19">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><a NAME="E8:33_1_1_6"/><span class="lab"><font color="Green" title="E39">A52</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a>, <a class="txt" href="#E5:33_1_1_6"><span class="lab"><font color="Green" title="E37">A50</font></span></a></span>;<br/><a NAME="E9:33_1_1_6"/>
<font color="Maroon" title="c20">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c16">fk</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a>, <a class="txt" href="#E5:33_1_1_6"><span class="lab"><font color="Green" title="E37">A50</font></span></a></span>;<br/><a NAME="E10:33_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A53</font></span>: 
<font color="Maroon" title="c20">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">fk</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#D4" onmouseover="rs('ltlaxio2/D4')" onmouseout="rh()">LTLAXIO2:def 4</a></span>;<br/><span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c16">fk</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i</font> = 
<span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c16">fk</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33_1_1_6"><span class="lab"><font color="Green" title="E37">A50</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#D4" onmouseover="rs('ltlaxio2/D4')" onmouseout="rh()">LTLAXIO2:def 4</a>, <a class="txt" href="#E10:33_1_1_6"><span class="lab"><font color="Green" title="E40">A53</font></span></a>, <a class="txt" href="#E8:33_1_1_6"><span class="lab"><font color="Green" title="E39">A52</font></span></a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A54</font></span>: <span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c16">fk</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span></span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="ltlaxio1.html#NK2" title="LTLAXIO1:NK.2">TFALSUM</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D15" onmouseover="rs('ltlaxio1/D15')" onmouseout="rh()">LTLAXIO1:def 15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span></span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D15" onmouseover="rs('ltlaxio1/D15')" onmouseout="rh()">LTLAXIO1:def 15</a></span>
;<br/><span class="lab"><font color="Green" title="E42">A55</font></span>: <span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:33_1_1_6"><span class="lab"><font color="Green" title="E40">A53</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a>, <a class="txt" href="#E8:33_1_1_6"><span class="lab"><font color="Green" title="E39">A52</font></span></a>, <a class="txt" href="#E25:33_1_1"><span class="lab"><font color="Green" title="E35">A48</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a> )</span>)</span> <a href="margrel1.html#K9" title="MARGREL1:func.9">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#T31" onmouseover="rs('ltlaxio1/T31')" onmouseout="rh()">LTLAXIO1:31</a></span>
;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_4"><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">  not <span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span></a><div class="add"><a NAME="E1:33_1_1_6_4"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c13">ff1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c13">ff1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T26" onmouseover="rs('finseq_1/T26')" onmouseout="rh()">FINSEQ_1:26</a></span>;<br/><a NAME="E2:33_1_1_6_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A56</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c13">ff1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T16" onmouseover="rs('ltlaxio2/T16')" onmouseout="rh()">LTLAXIO2:16</a></span>;<br/><span class="kw">assume </span><a NAME="E3:33_1_1_6_4"/>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> contradiction</span><br/><a NAME="E4:33_1_1_6_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span>)</span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="ltlaxio1.html#NK2" title="LTLAXIO1:NK.2">TFALSUM</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D15" onmouseover="rs('ltlaxio1/D15')" onmouseout="rh()">LTLAXIO1:def 15</a></span>;<br/><a NAME="E5:33_1_1_6_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A57</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span>)</span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D15" onmouseover="rs('ltlaxio1/D15')" onmouseout="rh()">LTLAXIO1:def 15</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_6_4_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="E1:33_1_1_6_4_1"/>
( <span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboolean.html#D3" onmouseover="rs('xboolean/D3')" onmouseout="rh()">XBOOLEAN:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_6_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:33_1_1_6_4_1_1"/><span class="lab"><font color="Green" title="E45">A58</font></span>: 
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.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"> contradiction</span><br/><div class="add"><a NAME="E2:33_1_1_6_4_1_1"/>
<font color="Maroon" title="c20">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c13">ff1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:33_1_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E25:33_1_1"><span class="lab"><font color="Green" title="E35">A48</font></span></a>, <a class="txt" href="#E10:33_1_1_6"><span class="lab"><font color="Green" title="E40">A53</font></span></a>, <a class="ref" href="ltlaxio2.html#D4" onmouseover="rs('ltlaxio2/D4')" onmouseout="rh()">LTLAXIO2:def 4</a></span>;<br/><a NAME="E3:33_1_1_6_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A59</font></span>: 
<font color="Maroon" title="c19">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c13">ff1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33_1_1_6"><span class="lab"><font color="Green" title="E39">A52</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c21">a</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c22">b</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a> ;<br/><a NAME="E4:33_1_1_6_4_1_1"/>
<font color="Maroon" title="c20">i1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">ff1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33_1_1_6"><span class="lab"><font color="Green" title="E39">A52</font></span></a>, <a class="txt" href="#E11:33_1_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E25:33_1_1"><span class="lab"><font color="Green" title="E35">A48</font></span></a>, <a class="txt" href="#E10:33_1_1_6"><span class="lab"><font color="Green" title="E40">A53</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E5:33_1_1_6_4_1_1"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:33_1_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a></span>;<br/><a NAME="E6:33_1_1_6_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A60</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio3.html#D3" onmouseover="rs('ltlaxio3/D3')" onmouseout="rh()">LTLAXIO3:def 3</a></span>;<br/><span class="lab"><font color="Green" title="E48">A61</font></span>: <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c13">ff1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T16" onmouseover="rs('ltlaxio2/T16')" onmouseout="rh()">LTLAXIO2:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c13">ff1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a>, <a class="txt" href="#E3:33_1_1_6_4_1_1"><span class="lab"><font color="Green" title="E46">A59</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c13">ff1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a>, <a class="txt" href="#E3:33_1_1_6_4_1_1"><span class="lab"><font color="Green" title="E46">A59</font></span></a></span>
<br/>.= 
 <a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#D4" onmouseover="rs('ltlaxio2/D4')" onmouseout="rh()">LTLAXIO2:def 4</a>, <a class="txt" href="#E8:33_1_1_6"><span class="lab"><font color="Green" title="E39">A52</font></span></a>, <a class="txt" href="#E11:33_1_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E25:33_1_1"><span class="lab"><font color="Green" title="E35">A48</font></span></a>, <a class="txt" href="#E10:33_1_1_6"><span class="lab"><font color="Green" title="E40">A53</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E49">A62</font></span>: 1 = 
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33_1_1_6_4"><span class="lab"><font color="Green" title="E44">A57</font></span></a>, <a class="txt" href="#E3:33_1_1_6_4_1_1"><span class="lab"><font color="Green" title="E46">A59</font></span></a>, <a class="txt" href="#E2:33_1_1_6_4"><span class="lab"><font color="Green" title="E43">A56</font></span></a>, <a class="ref" href="ltlaxio2.html#T19" onmouseover="rs('ltlaxio2/T19')" onmouseout="rh()">LTLAXIO2:19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span></span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="ltlaxio1.html#NK2" title="LTLAXIO1:NK.2">TFALSUM</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D15" onmouseover="rs('ltlaxio1/D15')" onmouseout="rh()">LTLAXIO1:def 15</a>, <a class="txt" href="#E7:33_1_1_6_4_1_1"><span class="lab"><font color="Green" title="E48">A61</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span></span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D15" onmouseover="rs('ltlaxio1/D15')" onmouseout="rh()">LTLAXIO1:def 15</a></span>
;<br/><span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">]</a></span> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:33_1_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E8:33_1_1_6"><span class="lab"><font color="Green" title="E39">A52</font></span></a>, <a class="txt" href="#E25:33_1_1"><span class="lab"><font color="Green" title="E35">A48</font></span></a>, <a class="txt" href="#E10:33_1_1_6"><span class="lab"><font color="Green" title="E40">A53</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>)</span>)</span> <a href="margrel1.html#K9" title="MARGREL1:func.9">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#T31" onmouseover="rs('ltlaxio1/T31')" onmouseout="rh()">LTLAXIO1:31</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a> )</span>)</span> <a href="margrel1.html#K9" title="MARGREL1:func.9">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>2</sub></font>(<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>)</span>)</span></span>)</span> <a href="margrel1.html#K9" title="MARGREL1:func.9">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T17" onmouseover="rs('ltlaxio2/T17')" onmouseout="rh()">LTLAXIO2:17</a>, <a class="txt" href="#E6:33_1_1_6_4_1_1"><span class="lab"><font color="Green" title="E47">A60</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a> )</span>)</span> <a href="margrel1.html#K9" title="MARGREL1:func.9">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">A</font></span>)</span></span>)</span> <a href="margrel1.html#K9" title="MARGREL1:func.9">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T11" onmouseover="rs('ltlaxio2/T11')" onmouseout="rh()">LTLAXIO2:11</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:33_1_1_6"><span class="lab"><font color="Green" title="E41">A54</font></span></a>, <a class="txt" href="#E6:33_1_1_6"><span class="lab"><font color="Green" title="E38">A51</font></span></a>, <a class="ref" href="xboolean.html#D3" onmouseover="rs('xboolean/D3')" onmouseout="rh()">XBOOLEAN:def 3</a>, <a class="txt" href="#E1:33_1_1_6_4_1_1"><span class="lab"><font color="Green" title="E45">A58</font></span></a>, <a class="txt" href="#E13:33_1_1_6"><span class="lab"><font color="Green" title="E42">A55</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E10:33_1_1_6_4_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33_1_1_6_4_1_1"><span class="lab"><font color="Green" title="E49">A62</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="33_1_1_6_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:33_1_1_6_4_1_2"/><span class="lab"><font color="Green" title="E45">A63</font></span>: 
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> contradiction</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c21">b</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c22">a</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a> ;<br/><a NAME="E2:33_1_1_6_4_1_2"/>
<font color="Maroon" title="c20">i1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">ff2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33_1_1_6"><span class="lab"><font color="Green" title="E39">A52</font></span></a>, <a class="txt" href="#E10:33_1_1_6"><span class="lab"><font color="Green" title="E40">A53</font></span></a>, <a class="txt" href="#E13:33_1_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E25:33_1_1"><span class="lab"><font color="Green" title="E35">A48</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E3:33_1_1_6_4_1_2"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:33_1_1"><span class="lab"><font color="Green" title="E31">A15</font></span></a></span>;<br/><a NAME="E4:33_1_1_6_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A64</font></span>: 
(  <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T16" onmouseover="rs('ltlaxio2/T16')" onmouseout="rh()">LTLAXIO2:16</a>, <a class="ref" href="ltlaxio3.html#D3" onmouseover="rs('ltlaxio3/D3')" onmouseout="rh()">LTLAXIO3:def 3</a></span>;<br/><a NAME="E5:33_1_1_6_4_1_2"/>
 <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <font color="Maroon" title="c9">A</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T14" onmouseover="rs('ltlaxio2/T14')" onmouseout="rh()">LTLAXIO2:14</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>) = 
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <font color="Maroon" title="c9">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T11" onmouseover="rs('ltlaxio2/T11')" onmouseout="rh()">LTLAXIO2:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="ltlaxio1.html#NK2" title="LTLAXIO1:NK.2">TFALSUM</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D15" onmouseover="rs('ltlaxio1/D15')" onmouseout="rh()">LTLAXIO1:def 15</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1_1_6_4_1_2"><span class="lab"><font color="Green" title="E45">A63</font></span></a></span>
;<br/><a NAME="E7:33_1_1_6_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A65</font></span>: 
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span>)</span>)</span> <a href="margrel1.html#K9" title="MARGREL1:func.9">'&amp;'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1_1_6_4_1_2"><span class="lab"><font color="Green" title="E46">A64</font></span></a>, <a class="ref" href="ltlaxio2.html#T17" onmouseover="rs('ltlaxio2/T17')" onmouseout="rh()">LTLAXIO2:17</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">j</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">ff1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c20">i1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/><a NAME="E9:33_1_1_6_4_1_2"/><span class="lab"><font color="Green" title="E48">A66</font></span>: 
<font color="Maroon" title="c23">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c13">ff1</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c20">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#D4" onmouseover="rs('ltlaxio2/D4')" onmouseout="rh()">LTLAXIO2:def 4</a></span>;<br/><a NAME="E10:33_1_1_6_4_1_2"/>
<font color="Maroon" title="c20">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c14">ff2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:33_1_1_6"><span class="lab"><font color="Green" title="E40">A53</font></span></a>, <a class="txt" href="#E13:33_1_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E25:33_1_1"><span class="lab"><font color="Green" title="E35">A48</font></span></a>, <a class="ref" href="ltlaxio2.html#D4" onmouseover="rs('ltlaxio2/D4')" onmouseout="rh()">LTLAXIO2:def 4</a></span>;<br/><a NAME="E11:33_1_1_6_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A67</font></span>: 
<font color="Maroon" title="c20">i1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c14">ff2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a>, <a class="txt" href="#E8:33_1_1_6"><span class="lab"><font color="Green" title="E39">A52</font></span></a></span>;<br/><a NAME="E12:33_1_1_6_4_1_2"/><span class="lab"><font color="Green" title="E50">A68</font></span>: 
<font color="Maroon" title="c23">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T28" onmouseover="rs('finseq_1/T28')" onmouseout="rh()">FINSEQ_1:28</a>, <a class="txt" href="#E8:33_1_1_6"><span class="lab"><font color="Green" title="E39">A52</font></span></a>, <a class="txt" href="#E10:33_1_1_6"><span class="lab"><font color="Green" title="E40">A53</font></span></a>, <a class="txt" href="#E13:33_1_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E25:33_1_1"><span class="lab"><font color="Green" title="E35">A48</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E13:33_1_1_6_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c23">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E14:33_1_1_6_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A69</font></span>: 
<font color="Maroon" title="c23">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#D4" onmouseover="rs('ltlaxio2/D4')" onmouseout="rh()">LTLAXIO2:def 4</a></span>;<br/><span class="lab"><font color="Green" title="E52">A70</font></span>: <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c13">ff1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">j</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T16" onmouseover="rs('ltlaxio2/T16')" onmouseout="rh()">LTLAXIO2:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c14">ff2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T69" onmouseover="rs('finseq_4/T69')" onmouseout="rh()">FINSEQ_4:69</a>, <a class="txt" href="#E11:33_1_1_6_4_1_2"><span class="lab"><font color="Green" title="E49">A67</font></span></a>, <a class="txt" href="#E9:33_1_1_6_4_1_2"><span class="lab"><font color="Green" title="E48">A66</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c14">ff2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a>, <a class="txt" href="#E11:33_1_1_6_4_1_2"><span class="lab"><font color="Green" title="E49">A67</font></span></a></span>
<br/>.= 
 <a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">ff2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#D4" onmouseover="rs('ltlaxio2/D4')" onmouseout="rh()">LTLAXIO2:def 4</a>, <a class="txt" href="#E8:33_1_1_6"><span class="lab"><font color="Green" title="E39">A52</font></span></a>, <a class="txt" href="#E10:33_1_1_6"><span class="lab"><font color="Green" title="E40">A53</font></span></a>, <a class="txt" href="#E13:33_1_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E25:33_1_1"><span class="lab"><font color="Green" title="E35">A48</font></span></a></span>
;<br/><a NAME="E16:33_1_1_6_4_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:33_1_1_6_4_1_2"><span class="lab"><font color="Green" title="E50">A68</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E17:33_1_1_6_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c23">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:33_1_1_6_4_1_2"><span class="lab"><font color="Green" title="E51">A69</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A71</font></span>: 1 = 
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33_1_1_6_4"><span class="lab"><font color="Green" title="E44">A57</font></span></a>, <a class="ref" href="ltlaxio2.html#T19" onmouseover="rs('ltlaxio2/T19')" onmouseout="rh()">LTLAXIO2:19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">ff2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span></span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="ltlaxio1.html#NK2" title="LTLAXIO1:NK.2">TFALSUM</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D15" onmouseover="rs('ltlaxio1/D15')" onmouseout="rh()">LTLAXIO1:def 15</a>, <a class="txt" href="#E15:33_1_1_6_4_1_2"><span class="lab"><font color="Green" title="E52">A70</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">ff2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span></span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D15" onmouseover="rs('ltlaxio1/D15')" onmouseout="rh()">LTLAXIO1:def 15</a></span>
;<br/><span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">ff2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">]</a></span> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:33_1_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E8:33_1_1_6"><span class="lab"><font color="Green" title="E39">A52</font></span></a>, <a class="txt" href="#E10:33_1_1_6"><span class="lab"><font color="Green" title="E40">A53</font></span></a>, <a class="txt" href="#E25:33_1_1"><span class="lab"><font color="Green" title="E35">A48</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a> )</span>)</span> <a href="margrel1.html#K9" title="MARGREL1:func.9">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i1</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="ltlaxio3.html#K3" title="LTLAXIO3:func.3">^^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#T31" onmouseover="rs('ltlaxio1/T31')" onmouseout="rh()">LTLAXIO1:31</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:33_1_1_6"><span class="lab"><font color="Green" title="E41">A54</font></span></a>, <a class="txt" href="#E6:33_1_1_6"><span class="lab"><font color="Green" title="E38">A51</font></span></a>, <a class="ref" href="xboolean.html#D3" onmouseover="rs('xboolean/D3')" onmouseout="rh()">XBOOLEAN:def 3</a>, <a class="txt" href="#E13:33_1_1_6"><span class="lab"><font color="Green" title="E42">A55</font></span></a>, <a class="txt" href="#E7:33_1_1_6_4_1_2"><span class="lab"><font color="Green" title="E47">A65</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E20:33_1_1_6_4_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:33_1_1_6_4_1_2"><span class="lab"><font color="Green" title="E53">A71</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="E15:33_1_1_6"/>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboolean.html#D3" onmouseover="rs('xboolean/D3')" onmouseout="rh()">XBOOLEAN:def 3</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="E27:33_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">fk</font>) <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>) is  <a href="ltlaxio2.html#NV1" title="LTLAXIO2:NV.1">ctaut</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>,<a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ltlaxio1.html#D16" onmouseover="rs('ltlaxio1/D16')" onmouseout="rh()">LTLAXIO1:def 16</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"> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">fk</font>) <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<span class="kw">set </span><font color="Maroon" title="c18">v</font> =  <a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_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">  not <span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">fk</font>) <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:33_1_1_7_1"/>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">fk</font>) <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> contradiction</span><br/><a NAME="E2:33_1_1_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A72</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">fk</font>)</span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D15" onmouseover="rs('ltlaxio1/D15')" onmouseout="rh()">LTLAXIO1:def 15</a></span>;<br/><a NAME="E3:33_1_1_7_1"/>
( <span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboolean.html#D3" onmouseover="rs('xboolean/D3')" onmouseout="rh()">XBOOLEAN:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:33_1_1_7_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_1_1_7_1"><span class="lab"><font color="Green" title="E37">A72</font></span></a>, <a class="txt" href="#E26:33_1_1"><span class="lab"><font color="Green" title="E36">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">hence </span><a NAME="E2:33_1_1_7"/>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">fk</font>) <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboolean.html#D3" onmouseover="rs('xboolean/D3')" onmouseout="rh()">XBOOLEAN:def 3</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="E28:33_1_1"/><span class="kw">then </span>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">fk</font>) <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio1.html#K13" title="LTLAXIO1:func.13">LTL_axioms</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D17" onmouseover="rs('ltlaxio1/D17')" onmouseout="rh()">LTLAXIO1:def 17</a></span>;<br/>
<a NAME="E29:33_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A73</font></span>: 
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">fk</font>) <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#T42" onmouseover="rs('ltlaxio1/T42')" onmouseout="rh()">LTLAXIO1:42</a></span>;<br/>
<a NAME="E30:33_1_1"/><span class="lab"><font color="Green" title="E38">A74</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c16">fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_1_8"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <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="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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"> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c16">fk</font></span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c17">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="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c16">fk</font> implies <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:33_1_1_8_1"/><span class="lab"><font color="Green" title="E39">A75</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c16">fk</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="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a> </span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">x1</font> = <font color="Maroon" title="c17">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>  ;<br/><span class="kw">set </span><font color="Maroon" title="c19">i</font> = <font color="Maroon" title="c18">x1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c16">fk</font>;<br/><a NAME="E3:33_1_1_8_1"/>
<font color="Maroon" title="c18">x1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c16">fk</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">fk</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1_1_8_1"><span class="lab"><font color="Green" title="E39">A75</font></span></a>, <a class="ref" href="finseq_4.html#T20" onmouseover="rs('finseq_4/T20')" onmouseout="rh()">FINSEQ_4:20</a></span>;<br/><a NAME="E4:33_1_1_8_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">x1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c16">fk</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">ff</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:33_1_1"><span class="lab"><font color="Green" title="E35">A48</font></span></a>, <a class="ref" href="finseq_3.html#T29" onmouseover="rs('finseq_3/T29')" onmouseout="rh()">FINSEQ_3:29</a></span>;<br/><a NAME="E5:33_1_1_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A76</font></span>: 
<font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c16">fk</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33_1_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/><font color="Maroon" title="c18">x1</font> = 
<font color="Maroon" title="c16">fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c16">fk</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1_1_8_1"><span class="lab"><font color="Green" title="E39">A75</font></span></a>, <a class="ref" href="finseq_5.html#T38" onmouseover="rs('finseq_5/T38')" onmouseout="rh()">FINSEQ_5:38</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c16">fk</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:33_1_1"><span class="lab"><font color="Green" title="E35">A48</font></span></a>, <a class="txt" href="#E1:33_1_1_8_1"><span class="lab"><font color="Green" title="E39">A75</font></span></a>, <a class="ref" href="finseq_4.html#T20" onmouseover="rs('finseq_4/T20')" onmouseout="rh()">FINSEQ_4:20</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:33_1_1_8_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33_1_1_8_1"><span class="lab"><font color="Green" title="E40">A76</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">let </span><font color="Maroon" title="c17">x</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="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a>  or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c16">fk</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:33_1_1_8"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.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"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c16">fk</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">P1</font> being   <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</a><span class="kw"> such that </span><br/><a NAME="E4:33_1_1_8"/><span class="lab"><font color="Green" title="E39">A77</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">P1</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a> 
 <span class="kw">and </span><br/><a NAME="E5:33_1_1_8"/><span class="lab"><font color="Green" title="E40">A78</font></span>: 
<font color="Maroon" title="c18">P1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 ;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">i</font> = <font color="Maroon" title="c18">P1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c12">ff</font>;<br/>
<a NAME="E6:33_1_1_8"/>
<font color="Maroon" title="c18">P1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c12">ff</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">ff</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T20" onmouseover="rs('finseq_4/T20')" onmouseout="rh()">FINSEQ_4:20</a>, <a class="txt" href="#E7:33_1_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E5:33_1_1_8"><span class="lab"><font color="Green" title="E40">A78</font></span></a></span>;<br/>
<a NAME="E7:33_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A79</font></span>: 
<font color="Maroon" title="c18">P1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c12">ff</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">fk</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" onmouseover="rs('finseq_3/T29')" onmouseout="rh()">FINSEQ_3:29</a>, <a class="txt" href="#E25:33_1_1"><span class="lab"><font color="Green" title="E35">A48</font></span></a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c16">fk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">P1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c12">ff</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">ff</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">P1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c12">ff</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:33_1_1"><span class="lab"><font color="Green" title="E35">A48</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c17">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1_1_8"><span class="lab"><font color="Green" title="E39">A77</font></span></a>, <a class="ref" href="finseq_5.html#T38" onmouseover="rs('finseq_5/T38')" onmouseout="rh()">FINSEQ_5:38</a>, <a class="txt" href="#E7:33_1_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E5:33_1_1_8"><span class="lab"><font color="Green" title="E40">A78</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:33_1_1_8"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c16">fk</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a>, <a class="txt" href="#E7:33_1_1_8"><span class="lab"><font color="Green" title="E41">A79</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="E31:33_1_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="stirl2_1.html#T55" onmouseover="rs('stirl2_1/T55')" onmouseout="rh()">STIRL2_1:55</a>, <a class="txt" href="#E4:33_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E5:33_1_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E1:33_1_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a></span>;<br/>
<a NAME="E32:33_1_1"/><span class="kw">then </span>
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">fk</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:33_1_1"><span class="lab"><font color="Green" title="E38">A74</font></span></a>, <a class="txt" href="#E1:33_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a></span>;<br/>
<a NAME="E33:33_1_1"/><span class="kw">then </span>
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">ff1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">ff2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:33_1_1"><span class="lab"><font color="Green" title="E37">A73</font></span></a>, <a class="ref" href="ltlaxio1.html#T43" onmouseover="rs('ltlaxio1/T43')" onmouseout="rh()">LTLAXIO1:43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E34:33_1_1"/>
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">g</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:33_1_1"><span class="lab"><font color="Green" title="E34">A47</font></span></a>, <a class="ref" href="ltlaxio1.html#T43" onmouseover="rs('ltlaxio1/T43')" onmouseout="rh()">LTLAXIO1:43</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>
<a NAME="E4:33"/><span class="lab"><font color="Green" title="E21">A80</font></span>: 
<font color="Maroon">S<sub>1</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="33_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">F</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</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"> (  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  for <font color="Olive" title="b1">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>   st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a>  holds <br/> <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">f</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_2"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">  for <font color="Olive" title="b1">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>   st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a>  holds <br/> <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">f</font>)</span><br/>

<a NAME="E2:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A81</font></span>: 
<font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>
 
;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</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"> (  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a>  implies  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">f</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E3:33_2"/><span class="lab"><font color="Green" title="E23">A82</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.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">  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">f</font>)</span><br/>

<a NAME="E4:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A83</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#T42" onmouseover="rs('finseq_6/T42')" onmouseout="rh()">FINSEQ_6:42</a>, <a class="ref" href="relat_1.html#T38" onmouseover="rs('relat_1/T38')" onmouseout="rh()">RELAT_1:38</a></span>;<br/>
<a NAME="E5:33_2"/><span class="lab"><font color="Green" title="E25">A84</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_2"><span class="lab"><font color="Green" title="E23">A82</font></span></a>, <a class="ref" href="finseq_3.html#T32" onmouseover="rs('finseq_3/T32')" onmouseout="rh()">FINSEQ_3:32</a></span>;<br/>
<a NAME="E6:33_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">f</font>) is  <a href="ltlaxio2.html#NV1" title="LTLAXIO2:NV.1">ctaut</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>,<a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ltlaxio1.html#D16" onmouseover="rs('ltlaxio1/D16')" onmouseout="rh()">LTLAXIO1:def 16</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"> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">v</font> =  <a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c7">g</font>;<br/>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K3" title="LTLAXIO1:func.3">TVERUM</a> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <a href="ltlaxio1.html#K3" title="LTLAXIO1:func.3">TVERUM</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="ref" href="ltlaxio4.html#T11" target="_self" onmouseover="rs('ltlaxio4/T11')" onmouseout="rh()">Th11</a>, <a class="txt" href="#E3:33_2"><span class="lab"><font color="Green" title="E23">A82</font></span></a>, <a class="ref" href="ltlaxio4.html#T12" target="_self" onmouseover="rs('ltlaxio4/T12')" onmouseout="rh()">Th12</a>, <a class="txt" href="#E2:33_2"><span class="lab"><font color="Green" title="E22">A81</font></span></a>, <a class="txt" href="#E4:33_2"><span class="lab"><font color="Green" title="E24">A83</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="ltlaxio1.html#K3" title="LTLAXIO1:func.3">TVERUM</a></span>)</span> <a href="margrel1.html#K9" title="MARGREL1:func.9">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="ltlaxio1.html#K3" title="LTLAXIO1:func.3">TVERUM</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#T31" onmouseover="rs('ltlaxio1/T31')" onmouseout="rh()">LTLAXIO1:31</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T4" onmouseover="rs('ltlaxio2/T4')" onmouseout="rh()">LTLAXIO2:4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:33_2_1"/>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboolean.html#D3" onmouseover="rs('xboolean/D3')" onmouseout="rh()">XBOOLEAN:def 3</a>, <a class="ref" href="ltlaxio2.html#T20" onmouseover="rs('ltlaxio2/T20')" onmouseout="rh()">LTLAXIO2:20</a>, <a class="txt" href="#E5:33_2"><span class="lab"><font color="Green" title="E25">A84</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="E7:33_2"/><span class="kw">then </span>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">f</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio1.html#K13" title="LTLAXIO1:func.13">LTL_axioms</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D17" onmouseover="rs('ltlaxio1/D17')" onmouseout="rh()">LTLAXIO1:def 17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:33_2"/>
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">f</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#T42" onmouseover="rs('ltlaxio1/T42')" onmouseout="rh()">LTLAXIO1:42</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="E5:33"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</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="#E4:33"><span class="lab"><font color="Green" title="E21">A80</font></span></a>, <a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E20">A3</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:33"/>
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">f</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E18">A1</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>
