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<span class="kw">let </span><font color="Maroon" title="c3">P</font> be   <a href="ltlaxio3.html#NV3" title="LTLAXIO3:NV.3">consistent</a>  <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</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#K5" title="LTLAXIO4:func.5">comp</a> <font color="Maroon" title="c3">P</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</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"><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> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="ltlaxio2.html#K2" title="LTLAXIO2:func.2">con</a> <span class="p5">(<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>)</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#K5" title="LTLAXIO4:func.5">comp</a> <font color="Maroon" title="c3">P</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</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"><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> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="ltlaxio2.html#K2" title="LTLAXIO2:func.2">con</a> <span class="p5">(<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>)</span> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c5">c</font> =  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</a>] <span class="kw">means </span>(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default">$1 <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default">$1 <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> );<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</a>] <span class="kw">means </span>( $1 is  <a href="ltlaxio3.html#NV3" title="LTLAXIO3:NV.3">consistent</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[$1] );<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</a>] <span class="kw">means </span>(  not  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</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">$1 <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> or  not  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <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">$1 <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> );<br/>
<span class="kw">set </span><font color="Maroon" title="c6">c1</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">Pg</font> where <font color="Olive" title="b1">Pg</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">Pg</font>] </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">c2</font> = <span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p2">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">Pg</font> where <font color="Olive" title="b1">Pg</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">Pg</font>] </span> } </span> ;<br/>
<a NAME="E1:34"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">Pg</font> where <font color="Olive" title="b1">Pg</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">Pg</font>] </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="fraenkel.html#S10" onmouseover="rs('fraenkel/S10')" onmouseout="rh()">FRAENKEL:sch 10</a>();<br/></span>
<a NAME="E2:34"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">Pg</font> where <font color="Olive" title="b1">Pg</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">Pg</font>] </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ltlaxio4.html#K5" title="LTLAXIO4:func.5">comp</a> <font color="Maroon" title="c3">P</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="34_1"><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">  <a href="ltlaxio4.html#K5" title="LTLAXIO4:func.5">comp</a> <font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">Pg</font> where <font color="Olive" title="b1">Pg</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">Pg</font>] </span> } </span> </span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c8">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="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">Pg</font> where <font color="Olive" title="b1">Pg</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">Pg</font>] </span> } </span>  implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ltlaxio4.html#K5" title="LTLAXIO4:func.5">comp</a> <font color="Maroon" title="c3">P</font> )</span><br/><span class="kw">assume </span><a NAME="E1:34_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">Pg</font> where <font color="Olive" title="b1">Pg</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">Pg</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="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ltlaxio4.html#K5" title="LTLAXIO4:func.5">comp</a> <font color="Maroon" title="c3">P</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">Pg</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:34_1_1"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<font color="Maroon" title="c9">Pg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">and </span><br/><a NAME="E4:34_1_1"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">Pg</font>]
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">Pg</font> = <font color="Maroon" title="c9">Pg</font> as   <a href="ltlaxio3.html#NV3" title="LTLAXIO3:NV.3">consistent</a>  <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_1_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a></span>;<br/><a NAME="E6:34_1_1"/>
<font color="Maroon" title="c10">Pg</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"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span>
 ;<br/><a NAME="E7:34_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">Q</font> being   <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</a> st <br/>( <font color="Olive" title="b1">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Pg</font> &amp;  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Olive" title="b1">Q</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"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></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">Q</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">Q</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> )
 ;<br/><a NAME="E8:34_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">Pg</font> <a href="ltlaxio4.html#R1" title="LTLAXIO4:pred.1">is_completion_of</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_1_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:34_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ltlaxio4.html#K5" title="LTLAXIO4:func.5">comp</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_1_1"><span class="lab"><font color="Green" title="E21">A3</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="c8">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="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ltlaxio4.html#K5" title="LTLAXIO4:func.5">comp</a> <font color="Maroon" title="c3">P</font> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">Pg</font> where <font color="Olive" title="b1">Pg</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">Pg</font>] </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:34_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ltlaxio4.html#K5" title="LTLAXIO4:func.5">comp</a> <font color="Maroon" title="c3">P</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">Pg</font> where <font color="Olive" title="b1">Pg</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">Pg</font>] </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">Q</font> being   <a href="ltlaxio3.html#NV3" title="LTLAXIO3:NV.3">consistent</a>  <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</a><span class="kw"> such that </span><br/><a NAME="E4:34_1"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
<font color="Maroon" title="c9">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">and </span><br/><a NAME="E5:34_1"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
<font color="Maroon" title="c9">Q</font> <a href="ltlaxio4.html#R1" title="LTLAXIO4:pred.1">is_completion_of</a> <font color="Maroon" title="c3">P</font>
 ;<br/>
<a NAME="E6:34_1"/>
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Q</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="Maroon" title="c9">Q</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> &amp;  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c9">Q</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"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio3.html#T30" onmouseover="rs('ltlaxio3/T30')" onmouseout="rh()">LTLAXIO3:30</a>, <a class="txt" href="#E5:34_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a></span>;<br/>
<a NAME="E7:34_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">Q</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"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">Q</font> = <font color="Maroon" title="c9">Q</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> ;<br/>
<a NAME="E9:34_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">Q</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:34_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">Pg</font> where <font color="Olive" title="b1">Pg</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">Pg</font>] </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_1"><span class="lab"><font color="Green" title="E21">A5</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">reconsider </span><font color="Maroon" title="c8">c1</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">Pg</font> where <font color="Olive" title="b1">Pg</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">Pg</font>] </span> } </span>  as   <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 <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="#E1:34"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<a NAME="E4:34"/><span class="lab"><font color="Green" title="E21">A7</font></span>: 
 <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c8">c1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p3">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"> { <span class="default"> <font color="Olive" title="b1">Pg</font> where <font color="Olive" title="b1">Pg</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">Pg</font>] </span> } </span> </span>)</span>
 
<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="txt" href="#E1:34"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">f2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E6:34"/><span class="lab"><font color="Green" title="E22">A8</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p3">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"> { <span class="default"> <font color="Olive" title="b1">Pg</font> where <font color="Olive" title="b1">Pg</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">Pg</font>] </span> } </span> </span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a> 
 <span class="kw">and </span><br/><a NAME="E7:34"/>
<font color="Maroon" title="c9">f2</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="c10">f2</font> = <font color="Maroon" title="c9">f2</font> as    <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">by</span> <span class="lab"><a class="txt" href="#E6:34"><span class="lab"><font color="Green" title="E22">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">set </span><font color="Maroon" title="c11">r</font> = <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="c4">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">f2</font></span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c12">s</font> = <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c13">t</font> = <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font>);<br/>
<span class="kw">assume </span><a NAME="E9:34"/><span class="lab"><font color="Green" title="E23">A9</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#K5" title="LTLAXIO4:func.5">comp</a> <font color="Maroon" title="c3">P</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</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"><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> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="ltlaxio2.html#K2" title="LTLAXIO2:func.2">con</a> <span class="p5">(<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>)</span></span><br/>

<div><span class="lab"><font color="Green" title="E24">A10</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="34_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">x</font> being   <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</a>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI: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"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">Pg</font> where <font color="Olive" title="b2">Pg</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">Pg</font>] </span> } </span>  holds <br/> not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">x</font> be   <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</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="c14">x</font> <a href="tarski.html#R2" title="TARSKI: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"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">Pg</font> where <font color="Olive" title="b1">Pg</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">Pg</font>] </span> } </span>  implies  not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c14">x</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:34_2"/><span class="lab"><font color="Green" title="E25">A11</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="tarski.html#R2" title="TARSKI: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"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">Pg</font> where <font color="Olive" title="b1">Pg</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">Pg</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">  not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c14">x</font>]</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">x1</font> = <font color="Maroon" title="c14">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><span class="kw">assume </span><a NAME="E3:34_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c14">x</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E4:34_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">c1</font>
 ;<br/><span class="kw">hence </span><a NAME="E5:34_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</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="E25">A12</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="34_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">pi</font> being   <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</a>  st <font color="Olive" title="b1">pi</font> <a href="tarski.html#R2" title="TARSKI: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"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">Pg</font> where <font color="Olive" title="b2">Pg</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">Pg</font>] </span> } </span>  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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">pi</font> be   <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</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="c14">pi</font> <a href="tarski.html#R2" title="TARSKI: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"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">Pg</font> where <font color="Olive" title="b1">Pg</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">Pg</font>] </span> } </span>  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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:34_3"/><span class="lab"><font color="Green" title="E26">A13</font></span>: 
<font color="Maroon" title="c14">pi</font> <a href="tarski.html#R2" title="TARSKI: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"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">Pg</font> where <font color="Olive" title="b1">Pg</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">Pg</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">  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span><br/><a NAME="E2:34_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">pi</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"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E3:34_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A14</font></span>: 
 ex <font color="Olive" title="b1">Q</font> being   <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</a> st <br/>( <font color="Olive" title="b1">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">pi</font> &amp;  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Olive" title="b1">Q</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"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></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">Q</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">Q</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_3_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="E1:34_3_1"/>
( <font color="Maroon" title="c14">pi</font> is  <a href="ltlaxio3.html#V2" title="LTLAXIO3:attr.2">Inconsistent</a>  or  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">pi</font>] )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="txt" href="#E1:34_3"><span class="lab"><font color="Green" title="E26">A13</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_3_1_1"/><span class="lab"><font color="Green" title="E28">A15</font></span>: 
<font color="Maroon" title="c14">pi</font> is  <a href="ltlaxio3.html#V2" title="LTLAXIO3:attr.2">Inconsistent</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:34_3_1_1"/>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span>)</span> is  <a href="ltlaxio2.html#NV1" title="LTLAXIO2:NV.1">ctaut</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T33" onmouseover="rs('ltlaxio2/T33')" onmouseout="rh()">LTLAXIO2:33</a></span>;<br/><a NAME="E3:34_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span>)</span> <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="E4:34_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A16</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> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span>)</span>
 <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="E5:34_3_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>  <a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_3_1_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:34_3_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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_3_1_1"><span class="lab"><font color="Green" title="E29">A16</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_3_1_2"/><span class="lab"><font color="Green" title="E28">A17</font></span>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">pi</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="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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span><br/><div class="add"><div><span class="lab"><font color="Green" title="E29">A18</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="34_3_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c14">pi</font>]</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_3_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:34_3_1_2_1_1"/>
(  not  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> or  not  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_3_1_2"><span class="lab"><font color="Green" title="E28">A17</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_3_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_3_1_2_1_1_1"/>
 not  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</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"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c14">pi</font>]</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:34_3_1_2_1_1_1"/><span class="lab"><font color="Green" title="E30">A19</font></span>: 
<font color="Maroon" title="c15">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="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:34_3_1_2_1_1_1"/><span class="lab"><font color="Green" title="E31">A20</font></span>: 
 not <font color="Maroon" title="c15">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="c14">pi</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>
 ;<br/><a NAME="E5:34_3_1_2_1_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font>
 <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></span>;<br/><a NAME="E6:34_3_1_2_1_1_1"/><span class="kw">then </span>
(  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> &amp; <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio3.html#T16" onmouseover="rs('ltlaxio3/T16')" onmouseout="rh()">LTLAXIO3:16</a>, <a class="txt" href="#E3:34_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E30">A19</font></span></a></span>;<br/><a NAME="E7:34_3_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">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="c14">pi</font> <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="#E3:34_3"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a>, <a class="txt" href="#E4:34_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E31">A20</font></span></a></span>;<br/><a NAME="E8:34_3_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <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"><font color="Maroon" title="c14">pi</font> <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="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a>, <a class="txt" href="#E3:34_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E30">A19</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:34_3_1_2_1_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c14">pi</font>]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" onmouseover="rs('xboole_0/D7')" onmouseout="rh()">XBOOLE_0:def 7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_3_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_3_1_2_1_1_2"/>
 not  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">pi</font> <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"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c14">pi</font>]</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:34_3_1_2_1_1_2"/><span class="lab"><font color="Green" title="E30">A21</font></span>: 
<font color="Maroon" title="c15">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="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:34_3_1_2_1_1_2"/><span class="lab"><font color="Green" title="E31">A22</font></span>: 
 not <font color="Maroon" title="c15">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="c14">pi</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span>
 ;<br/><a NAME="E5:34_3_1_2_1_1_2"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font>
 <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></span>;<br/><a NAME="E6:34_3_1_2_1_1_2"/><span class="kw">then </span>
(  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> &amp; <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio3.html#T16" onmouseover="rs('ltlaxio3/T16')" onmouseout="rh()">LTLAXIO3:16</a>, <a class="txt" href="#E3:34_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E30">A21</font></span></a></span>;<br/><a NAME="E7:34_3_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">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="c14">pi</font> <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="#E3:34_3"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a>, <a class="txt" href="#E4:34_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E31">A22</font></span></a></span>;<br/><a NAME="E8:34_3_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <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"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a>, <a class="txt" href="#E3:34_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E30">A21</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:34_3_1_2_1_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c14">pi</font>]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" onmouseover="rs('xboole_0/D7')" onmouseout="rh()">XBOOLE_0:def 7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E30">A23</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="34_3_1_2_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">  <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"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_3_1_2_2_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="E1:34_3_1_2_2_1"/>
(  not  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</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="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> or  not  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <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"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_3_1_2"><span class="lab"><font color="Green" title="E29">A18</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_3_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_3_1_2_2_1_1"/><span class="lab"><font color="Green" title="E31">A24</font></span>: 
 not  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</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="Maroon" title="c14">pi</font> <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">  <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"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c15">Q1</font> = <span class="p1"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">pi</font> <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/><span class="kw">set </span><font color="Maroon" title="c16">p</font> = <font color="Maroon">H<sub>2</sub></font>(<span class="p1"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">pi</font> <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> );<br/><span class="kw">set </span><font color="Maroon" title="c17">q</font> = <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"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">pi</font> <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>);<br/><a NAME="E2:34_3_1_2_2_1_1"/>
 not  <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"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">pi</font> <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"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">pi</font> <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="txt" href="#E1:34_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E31">A24</font></span></a></span>;<br/><a NAME="E3:34_3_1_2_2_1_1"/><span class="kw">then </span>
<span class="p1"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">pi</font> <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> is  <a href="ltlaxio3.html#V2" title="LTLAXIO3:attr.2">Inconsistent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio3.html#T30" onmouseover="rs('ltlaxio3/T30')" onmouseout="rh()">LTLAXIO3:30</a></span>;<br/><a NAME="E4:34_3_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A25</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>  <a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</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"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">pi</font> <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>
 ;<br/><a NAME="E5:34_3_1_2_2_1_1"/><span class="lab"><font color="Green" title="E33">A26</font></span>: 
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a> ) <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</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="c14">pi</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span>)</span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</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="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span>)</span> is  <a href="ltlaxio2.html#NV1" title="LTLAXIO2:NV.1">ctaut</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T42" onmouseover="rs('ltlaxio2/T42')" onmouseout="rh()">LTLAXIO2:42</a></span>;<br/><a NAME="E6:34_3_1_2_2_1_1"/>
<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"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c14">pi</font> <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>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</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="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span>)</span> <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="txt" href="#E5:34_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E33">A26</font></span></a>, <a class="ref" href="ltlaxio1.html#D17" onmouseover="rs('ltlaxio1/D17')" onmouseout="rh()">LTLAXIO1:def 17</a></span>;<br/><a NAME="E7:34_3_1_2_2_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> <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"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c14">pi</font> <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>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</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="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span>)</span>
 <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">hence </span><a NAME="E8:34_3_1_2_2_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>  <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="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#T43" onmouseover="rs('ltlaxio1/T43')" onmouseout="rh()">LTLAXIO1:43</a>, <a class="txt" href="#E4:34_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E32">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="34_3_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_3_1_2_2_1_2"/><span class="lab"><font color="Green" title="E31">A27</font></span>: 
 not  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <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"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</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">  <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"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c15">Q2</font> = <span class="p1"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <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/><span class="kw">set </span><font color="Maroon" title="c16">p</font> = <font color="Maroon">H<sub>2</sub></font>(<span class="p1"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <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> );<br/><span class="kw">set </span><font color="Maroon" title="c17">q</font> = <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"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <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>);<br/><a NAME="E2:34_3_1_2_2_1_2"/>
 not  <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"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <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>  <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"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E31">A27</font></span></a></span>;<br/><a NAME="E3:34_3_1_2_2_1_2"/><span class="kw">then </span>
<span class="p1"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <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> is  <a href="ltlaxio3.html#V2" title="LTLAXIO3:attr.2">Inconsistent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio3.html#T30" onmouseover="rs('ltlaxio3/T30')" onmouseout="rh()">LTLAXIO3:30</a></span>;<br/><a NAME="E4:34_3_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A28</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>  <a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</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"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <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>
 ;<br/><a NAME="E5:34_3_1_2_2_1_2"/><span class="lab"><font color="Green" title="E33">A29</font></span>: 
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a> ) <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</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="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span>)</span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</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="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span>)</span> is  <a href="ltlaxio2.html#NV1" title="LTLAXIO2:NV.1">ctaut</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T41" onmouseover="rs('ltlaxio2/T41')" onmouseout="rh()">LTLAXIO2:41</a></span>;<br/><a NAME="E6:34_3_1_2_2_1_2"/>
<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"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">P</font> <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>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</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="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span>)</span> <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="txt" href="#E5:34_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E33">A29</font></span></a>, <a class="ref" href="ltlaxio1.html#D17" onmouseover="rs('ltlaxio1/D17')" onmouseout="rh()">LTLAXIO1:def 17</a></span>;<br/><a NAME="E7:34_3_1_2_2_1_2"/><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> <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"><a href="ltlaxio3.html#K11" title="LTLAXIO3:func.11">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">P</font> <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>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</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="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span>)</span>
 <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">hence </span><a NAME="E8:34_3_1_2_2_1_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>  <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="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#T43" onmouseover="rs('ltlaxio1/T43')" onmouseout="rh()">LTLAXIO1:43</a>, <a class="txt" href="#E4:34_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E32">A28</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><a NAME="E4:34_3_1_2"/>
<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="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span>)</span> is  <a href="ltlaxio2.html#NV1" title="LTLAXIO2:NV.1">ctaut</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T37" onmouseover="rs('ltlaxio2/T37')" onmouseout="rh()">LTLAXIO2:37</a></span>;<br/><a NAME="E5:34_3_1_2"/><span class="kw">then </span>
<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="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span>)</span> <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="E6:34_3_1_2"/><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> <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="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span></span>)</span>
 <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">hence </span><a NAME="E7:34_3_1_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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">pi</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#T43" onmouseover="rs('ltlaxio1/T43')" onmouseout="rh()">LTLAXIO1:43</a>, <a class="txt" href="#E3:34_3_1_2"><span class="lab"><font color="Green" title="E30">A23</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>
<a NAME="E12:34"/><span class="lab"><font color="Green" title="E26">A30</font></span>: 
 for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>   st <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p3">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"> { <span class="default"> <font color="Olive" title="b2">Pg</font> where <font color="Olive" title="b2">Pg</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">Pg</font>] </span> } </span> </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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <font color="Olive" title="b1">p</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="34_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">p</font> be    <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> ; <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="c14">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p3">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"> { <span class="default"> <font color="Olive" title="b1">Pg</font> where <font color="Olive" title="b1">Pg</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">Pg</font>] </span> } </span> </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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <font color="Maroon" title="c14">p</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:34_4"/>
<font color="Maroon" title="c14">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p3">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"> { <span class="default"> <font color="Olive" title="b1">Pg</font> where <font color="Olive" title="b1">Pg</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">Pg</font>] </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">  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <font color="Maroon" title="c14">p</font></span>)</span></span><br/>

<a NAME="E2:34_4"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">Q</font> being   <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</a> st <br/>( <font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Q</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a>  &amp; <font color="Olive" title="b1">Q</font> <a href="tarski.html#R2" title="TARSKI: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"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">Pg</font> where <font color="Olive" title="b2">Pg</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">Pg</font>] </span> } </span>  )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:34_4"/>
 <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <font color="Maroon" title="c14">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:34"><span class="lab"><font color="Green" title="E25">A12</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E27">A31</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="34_5"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_5_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="E1:34_5_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f2</font> <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> <font color="Maroon" title="c10">f2</font> <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="34_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_5_1_1"/><span class="lab"><font color="Green" title="E28">A32</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f2</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">  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font>)</span><br/><div class="add"><a NAME="E2:34_5_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <a href="ltlaxio1.html#K3" title="LTLAXIO1:func.3">TVERUM</a> 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="34_5_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">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="c14">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="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <a href="ltlaxio1.html#K3" title="LTLAXIO1:func.3">TVERUM</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<span class="kw">set </span><font color="Maroon" title="c15">v</font> =  <a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c14">g</font>;<br/>
<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="c14">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="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <a href="ltlaxio1.html#K3" title="LTLAXIO1:func.3">TVERUM</a></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="c14">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="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></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="c14">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#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="c14">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="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>
<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/>.= 
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><br/>


</div><span class="kw">end;</span></div><a NAME="E3:34_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A33</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <a href="ltlaxio1.html#K3" title="LTLAXIO1:func.3">TVERUM</a> <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="E4:34_5_1_1"/>
 <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="c10">f2</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="txt" href="#E1:34_5_1_1"><span class="lab"><font color="Green" title="E28">A32</font></span></a>, <a class="ref" href="ltlaxio2.html#D4" onmouseover="rs('ltlaxio2/D4')" onmouseout="rh()">LTLAXIO2:def 4</a></span>;<br/><a NAME="E5:34_5_1_1"/><span class="kw">then </span>
 <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><span class="kw">hence </span><a NAME="E6:34_5_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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_5_1_1"><span class="lab"><font color="Green" title="E29">A33</font></span></a>, <a class="ref" href="ltlaxio1.html#T42" onmouseover="rs('ltlaxio1/T42')" onmouseout="rh()">LTLAXIO1:42</a>, <a class="ref" href="ltlaxio2.html#T10" onmouseover="rs('ltlaxio2/T10')" onmouseout="rh()">LTLAXIO2:10</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="34_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_5_1_2"/><span class="lab"><font color="Green" title="E28">A34</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f2</font> <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">  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font>)</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c14">t1</font> =  <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> <font color="Maroon" title="c10">f2</font></span>)</span>;<br/><a NAME="E2:34_5_1_2"/><span class="lab"><font color="Green" title="E29">A35</font></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> <font color="Maroon" title="c10">f2</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_5_1_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="ref" href="ltlaxio2.html#D4" onmouseover="rs('ltlaxio2/D4')" onmouseout="rh()">LTLAXIO2:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">lt1</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K2" title="LTLAXIO2:func.2">con</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font></span>)</span></span>)</span> as   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <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">defpred </span><font color="Maroon">S<sub>4</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K2" title="LTLAXIO2:func.2">con</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font></span>)</span></span>)</span> 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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</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="c10">f2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span> );<br/><div><span class="lab"><font color="Green" title="E30">A36</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="34_5_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>4</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="c16">k</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <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>4</sub></font>[<font color="Maroon" title="c16">k</font>] implies <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:34_5_1_2_1"/><span class="lab"><font color="Green" title="E31">A37</font></span>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c16">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>4</sub></font>[<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><span class="kw">thus </span><a NAME="E2:34_5_1_2_1"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c16">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="34_5_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c17">a</font> = <span class="p1">(<span class="default"><a href="ltlaxio2.html#K2" title="LTLAXIO2:func.2">con</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">k</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">b</font> = <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">k1</font> = <font color="Maroon" title="c16">k</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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/>
<span class="kw">assume </span><a NAME="E2:34_5_1_2_1_1"/><span class="lab"><font color="Green" title="E32">A38</font></span>: 
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K2" title="LTLAXIO2:func.2">con</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</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="c10">f2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><br/>

<a NAME="E3:34_5_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><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="c10">f2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">k</font></span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ltlaxio2.html#K2" title="LTLAXIO2:func.2">con</a> <span class="p0">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> is  <a href="ltlaxio2.html#NV1" title="LTLAXIO2:NV.1">ctaut</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T40" onmouseover="rs('ltlaxio2/T40')" onmouseout="rh()">LTLAXIO2:40</a></span>;<br/>
<a NAME="E4:34_5_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><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="c10">f2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">k</font></span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ltlaxio2.html#K2" title="LTLAXIO2:func.2">con</a> <span class="p0">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <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="E5:34_5_1_2_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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><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="c10">f2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">k</font></span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ltlaxio2.html#K2" title="LTLAXIO2:func.2">con</a> <span class="p0">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<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="E6:34_5_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A39</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ltlaxio2.html#K2" title="LTLAXIO2:func.2">con</a> <span class="p5">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#T43" onmouseover="rs('ltlaxio1/T43')" onmouseout="rh()">LTLAXIO1:43</a>, <a class="txt" href="#E2:34_5_1_2_1_1"><span class="lab"><font color="Green" title="E32">A38</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a>, <a class="txt" href="#E1:34_5_1_2_1"><span class="lab"><font color="Green" title="E31">A37</font></span></a></span>;<br/>
<a NAME="E7:34_5_1_2_1_1"/><span class="lab"><font color="Green" title="E34">A40</font></span>: 
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_5_1_2_1_1"><span class="lab"><font color="Green" title="E32">A38</font></span></a>, <a class="ref" href="ltlaxio2.html#D2" onmouseover="rs('ltlaxio2/D2')" onmouseout="rh()">LTLAXIO2:def 2</a>, <a class="txt" href="#E2:34_5_1_2"><span class="lab"><font color="Green" title="E29">A35</font></span></a></span>;<br/>
<a NAME="E8:34_5_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f2</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="E9:34_5_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A41</font></span>: 
<font color="Maroon" title="c19">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 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="c10">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E10:34_5_1_2_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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</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="#E6:34"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E12:34"><span class="lab"><font color="Green" title="E26">A30</font></span></a></span>;<br/>
<a NAME="E11:34_5_1_2_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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</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="c10">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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="#E9:34_5_1_2_1_1"><span class="lab"><font color="Green" title="E35">A41</font></span></a></span>;<br/>
<a NAME="E12:34_5_1_2_1_1"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k1</font> &amp;  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="c10">f2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T25" onmouseover="rs('nat_1/T25')" onmouseout="rh()">NAT_1:25</a>, <a class="txt" href="#E6:34_5_1_2_1_1"><span class="lab"><font color="Green" title="E33">A39</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="E13:34_5_1_2_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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</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="c10">f2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T7" onmouseover="rs('ltlaxio2/T7')" onmouseout="rh()">LTLAXIO2:7</a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a>, <a class="txt" href="#E7:34_5_1_2_1_1"><span class="lab"><font color="Green" title="E34">A40</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><a NAME="E5:34_5_1_2"/><span class="lab"><font color="Green" title="E31">A42</font></span>: 
<font color="Maroon">S<sub>4</sub></font>[1]
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="34_5_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:34_5_1_2_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K2" title="LTLAXIO2:func.2">con</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</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="c10">f2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><br/>

<a NAME="E2:34_5_1_2_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_5_1_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="ref" href="nat_1.html#T25" onmouseover="rs('nat_1/T25')" onmouseout="rh()">NAT_1:25</a></span>;<br/>
<a NAME="E3:34_5_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A43</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="c10">f2</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/>
<span class="p1">(<span class="default"><a href="ltlaxio2.html#K2" title="LTLAXIO2:func.2">con</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_5_1_2"><span class="lab"><font color="Green" title="E29">A35</font></span></a>, <a class="ref" href="ltlaxio2.html#T6" onmouseover="rs('ltlaxio2/T6')" onmouseout="rh()">LTLAXIO2:6</a></span>
<br/>.= 
 <a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <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="ltlaxio2.html#T8" onmouseover="rs('ltlaxio2/T8')" onmouseout="rh()">LTLAXIO2:8</a>, <a class="txt" href="#E3:34_5_1_2_2"><span class="lab"><font color="Green" title="E32">A43</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:34_5_1_2_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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</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="c10">f2</font></span>)</span></span>)</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="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a>, <a class="txt" href="#E3:34_5_1_2_2"><span class="lab"><font color="Green" title="E32">A43</font></span></a>, <a class="txt" href="#E6:34"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E12:34"><span class="lab"><font color="Green" title="E26">A30</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="E6:34_5_1_2"/>
 for <font color="Olive" title="b1">k</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S10" onmouseover="rs('nat_1/S10')" onmouseout="rh()">NAT_1:sch 10</a>(<a class="txt" href="#E5:34_5_1_2"><span class="lab"><font color="Green" title="E31">A42</font></span></a>, <a class="txt" href="#E4:34_5_1_2"><span class="lab"><font color="Green" title="E30">A36</font></span></a>);<br/></span><a NAME="E7:34_5_1_2"/><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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</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="c10">f2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">lt1</font></span>)</span>
 ;<br/><span class="kw">hence </span><a NAME="E8:34_5_1_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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E14:34"/><span class="lab"><font color="Green" title="E28">A44</font></span>: 
 <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">f2</font></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> <font color="Maroon" title="c4">f</font></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> <font color="Maroon" title="c10">f2</font></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/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="34_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> holds  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</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="c4">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">f2</font></span>)</span>)</span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f</font>) <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font>)</span>)</span></span>)</span></span>)</span> <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="c14">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="c14">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="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</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="c4">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">f2</font></span>)</span>)</span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f</font>) <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font>)</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><span class="kw">set </span><font color="Maroon" title="c15">v</font> =  <a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c14">g</font>;<br/><a NAME="E1:34_6"/><span class="lab"><font color="Green" title="E29">A45</font></span>: 
( <span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c14">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="c4">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">f2</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a>  or <span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c14">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="c4">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">f2</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</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/><a NAME="E2:34_6"/><span class="lab"><font color="Green" title="E30">A46</font></span>: 
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c14">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="c4">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">f2</font></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="c14">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="c4">f</font>)</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="c14">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="c10">f2</font>)</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="#E14:34"><span class="lab"><font color="Green" title="E28">A44</font></span></a></span>;<br/><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="c14">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="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</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="c4">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">f2</font></span>)</span>)</span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f</font>) <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font>)</span>)</span></span>)</span></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="c14">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</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="c4">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">f2</font></span>)</span>)</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="c14">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f</font>) <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font>)</span>)</span></span>)</span></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"><span class="p3">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c14">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="c4">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">f2</font></span>)</span>)</span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</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="c14">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>)</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="c14">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f</font>) <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font>)</span>)</span></span>)</span></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"><span class="p3">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c14">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="c4">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">f2</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>)</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="c14">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f</font>) <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font>)</span>)</span></span>)</span></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"><span class="p3">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c14">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="c4">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">f2</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>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <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="c14">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f</font>) <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font>)</span>)</span></span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</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="c14">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>)</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"><span class="p3">(<span class="default"><a href="ltlaxio1.html#K12" title="LTLAXIO1:func.12">VAL</a> <font color="Maroon" title="c14">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="c4">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">f2</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>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <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="c14">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f</font>) <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</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>)</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="#E1:34_6"><span class="lab"><font color="Green" title="E29">A45</font></span></a>, <a class="txt" href="#E2:34_6"><span class="lab"><font color="Green" title="E30">A46</font></span></a>, <a class="ref" href="ltlaxio1.html#T31" onmouseover="rs('ltlaxio1/T31')" onmouseout="rh()">LTLAXIO1:31</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="E16:34"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</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="c4">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">f2</font></span>)</span>)</span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f</font>) <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font>)</span>)</span></span>)</span> is  <a href="ltlaxio2.html#NV1" title="LTLAXIO2:NV.1">ctaut</a> 
 
;<br/>
<a NAME="E17:34"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</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="c4">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">f2</font></span>)</span>)</span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f</font>) <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font>)</span>)</span></span>)</span> <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="E18:34"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">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> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</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="c4">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">f2</font></span>)</span>)</span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f</font>) <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font>)</span>)</span></span>)</span>
 
<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="E19:34"/>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f</font>) <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font>)</span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font>)</span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p3">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f</font>)</span>)</span></span>)</span></span>)</span> is  <a href="ltlaxio2.html#NV1" title="LTLAXIO2:NV.1">ctaut</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T39" onmouseover="rs('ltlaxio2/T39')" onmouseout="rh()">LTLAXIO2:39</a></span>;<br/>
<a NAME="E20:34"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f</font>) <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font>)</span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font>)</span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p3">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f</font>)</span>)</span></span>)</span></span>)</span> <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="E21:34"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A48</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> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f</font>) <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font>)</span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font>)</span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p3">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <font color="Maroon">H<sub>2</sub></font>( <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 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 href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">f2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">f</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="c10">f2</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/>.= 
<span class="p1">(<span class="default"><a href="ltlaxio4.html#K4" title="LTLAXIO4:func.4">comp</a> <span class="p2">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c3">P</font></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="#E4:34"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="ltlaxio4.html#T10" target="_self" onmouseover="rs('ltlaxio4/T10')" onmouseout="rh()">Th10</a>, <a class="txt" href="#E9:34"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E6:34"><span class="lab"><font color="Green" title="E22">A8</font></span></a></span>
;<br/>
<a NAME="E23:34"/><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>  <a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</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="c4">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">f2</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#T16" target="_self" onmouseover="rs('ltlaxio4/T16')" onmouseout="rh()">Th16</a></span>;<br/>
<a NAME="E24:34"/><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>  <a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f</font>) <a href="ltlaxio1.html#K4" title="LTLAXIO1:func.4">'&amp;&amp;'</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:34"><span class="lab"><font color="Green" title="E29">A47</font></span></a>, <a class="ref" href="ltlaxio1.html#T43" onmouseover="rs('ltlaxio1/T43')" onmouseout="rh()">LTLAXIO1:43</a></span>;<br/>
<a NAME="E25:34"/><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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c10">f2</font>)</span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:34"><span class="lab"><font color="Green" title="E30">A48</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="E26:34"/>
 <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#T43" onmouseover="rs('ltlaxio1/T43')" onmouseout="rh()">LTLAXIO1:43</a>, <a class="txt" href="#E13:34"><span class="lab"><font color="Green" title="E27">A31</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/>


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