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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#V3" title="LTLAXIO3:attr.3">complete</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">T</font> being    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c3">P</font><br/>  for <font color="Olive" title="b2">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>   st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Olive" title="b1">T</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"><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="b2">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="b2">f</font></span>)</span></span>)</span></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#K2" title="LTLAXIO1:func.2">'X'</a> <span class="p2">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <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="Olive" title="b2">f</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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="Olive" title="b2">f</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">T</font> be    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <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">  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#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Maroon" title="c4">T</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"><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> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K2" title="LTLAXIO1:func.2">'X'</a> <span class="p2">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <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="Olive" title="b1">f</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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="Olive" title="b1">f</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">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="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Maroon" title="c4">T</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"><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="c5">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="c5">f</font></span>)</span></span>)</span></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#K2" title="LTLAXIO1:func.2">'X'</a> <span class="p2">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <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="c5">f</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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="c5">f</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:63"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Maroon" title="c4">T</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"><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="c5">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="c5">f</font></span>)</span></span>)</span></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#K2" title="LTLAXIO1:func.2">'X'</a> <span class="p2">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <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="c5">f</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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="c5">f</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><br/>

<div><span class="lab"><font color="Green" title="E33">A2</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_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">R</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">T</font> holds   <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="Olive" title="b1">R</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#K2" title="LTLAXIO1:func.2">'X'</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>)</span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">R</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">T</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="c6">R</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#K2" title="LTLAXIO1:func.2">'X'</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>)</span>)</span></span><br/><span class="kw">consider </span><font color="Maroon" title="c7">Q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E2:63_1"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
<font color="Maroon" title="c7">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ltlaxio4.html#K8" title="LTLAXIO4:func.8">untn</a> <font color="Maroon" title="c6">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">Q</font> = <font color="Maroon" title="c7">Q</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K8" title="LTLAXIO4:func.8">untn</a> <font color="Maroon" title="c6">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1"><span class="lab"><font color="Green" title="E34">A3</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E5:63_1"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">g</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="c8">Q</font></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a> 
 <span class="kw">and </span><br/><a NAME="E6:63_1"/>
<font color="Maroon" title="c9">g</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">g</font> = <font color="Maroon" title="c9">g</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="#E5:63_1"><span class="lab"><font color="Green" title="E35">A4</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/><a NAME="E8:63_1"/><span class="lab"><font color="Green" title="E36">A5</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"><font color="Maroon" title="c6">R</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#K2" title="LTLAXIO1:func.2">'X'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">Q</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="ltlaxio4.html#T18" target="_self" onmouseover="rs('ltlaxio4/T18')" onmouseout="rh()">Th18</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">Q</font> = <font color="Maroon" title="c8">Q</font> as   <a href="ltlaxio3.html#NV3" title="LTLAXIO3:NV.3">consistent</a>  <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</a> ;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">g</font> 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="c10">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">j</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">g</font> 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="c10">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:63_1_1"/>
<font color="Maroon" title="c12">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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="c10">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>)</span><br/><a NAME="E2:63_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K5" title="LTLAXIO4:func.5">comp</a> <font color="Maroon" title="c11">Q</font></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a> 
 <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="#E5:63_1"><span class="lab"><font color="Green" title="E35">A4</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">S</font> being   <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</a><span class="kw"> such that </span><br/><a NAME="E4:63_1_1"/><span class="lab"><font color="Green" title="E37">A6</font></span>: 
<font color="Maroon" title="c10">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">S</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a> 
 <span class="kw">and </span><br/><a NAME="E5:63_1_1"/><span class="lab"><font color="Green" title="E38">A7</font></span>: 
<font color="Maroon" title="c13">S</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio4.html#K5" title="LTLAXIO4:func.5">comp</a> <font color="Maroon" title="c11">Q</font>
 ;<br/><a NAME="E6:63_1_1"/>
 <a href="ltlaxio4.html#K5" title="LTLAXIO4:func.5">comp</a> <font color="Maroon" title="c11">Q</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Maroon" title="c4">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#T28" target="_self" onmouseover="rs('ltlaxio4/T28')" onmouseout="rh()">Th28</a></span>;<br/><a NAME="E7:63_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">S</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:63_1_1"><span class="lab"><font color="Green" title="E38">A7</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">k</font> being    <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"> such that </span><br/><a NAME="E9:63_1_1"/><span class="lab"><font color="Green" title="E39">A8</font></span>: 
<font color="Maroon" title="c14">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E10:63_1_1"/><span class="lab"><font color="Green" title="E40">A9</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_1_1"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/><a NAME="E11:63_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>) 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#T29" onmouseover="rs('ltlaxio2/T29')" onmouseout="rh()">LTLAXIO2:29</a>, <a class="txt" href="#E9:63_1_1"><span class="lab"><font color="Green" title="E39">A8</font></span></a></span>;<br/><a NAME="E12:63_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio1.html#K13" title="LTLAXIO1:func.13">LTL_axioms</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D17" onmouseover="rs('ltlaxio1/D17')" onmouseout="rh()">LTLAXIO1:def 17</a></span>;<br/><span class="kw">hence </span><a NAME="E13:63_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="c10">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#T42" onmouseover="rs('ltlaxio1/T42')" onmouseout="rh()">LTLAXIO1:42</a>, <a class="txt" href="#E10:63_1_1"><span class="lab"><font color="Green" title="E40">A9</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E11:63_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A10</font></span>: 
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">g</font>) <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T57" onmouseover="rs('ltlaxio2/T57')" onmouseout="rh()">LTLAXIO2:57</a></span>;<br/><a NAME="E12:63_1"/>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K2" title="LTLAXIO1:func.2">'X'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">Q</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>1</sub></font>(<font color="Maroon" title="c5">f</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"><a href="ltlaxio1.html#K2" title="LTLAXIO1:func.2">'X'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">Q</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="p2">(<span class="default"><a href="ltlaxio1.html#K2" title="LTLAXIO1:func.2">'X'</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</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="E13:63_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A11</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#K2" title="LTLAXIO1:func.2">'X'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">Q</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>1</sub></font>(<font color="Maroon" title="c5">f</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"><a href="ltlaxio1.html#K2" title="LTLAXIO1:func.2">'X'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">Q</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="p2">(<span class="default"><a href="ltlaxio1.html#K2" title="LTLAXIO1:func.2">'X'</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</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="E14:63_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="c11">Q</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>1</sub></font>(<font color="Maroon" title="c10">g</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#T17" target="_self" onmouseover="rs('ltlaxio4/T17')" onmouseout="rh()">Th17</a>, <a class="txt" href="#E5:63_1"><span class="lab"><font color="Green" title="E35">A4</font></span></a></span>;<br/><a NAME="E15:63_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>  <a href="ltlaxio1.html#K2" title="LTLAXIO1:func.2">'X'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">Q</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>1</sub></font>(<font color="Maroon" title="c5">f</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:63_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="ref" href="ltlaxio1.html#T47" onmouseover="rs('ltlaxio1/T47')" onmouseout="rh()">LTLAXIO1:47</a>, <a class="ref" href="ltlaxio1.html#T44" onmouseover="rs('ltlaxio1/T44')" onmouseout="rh()">LTLAXIO1:44</a></span>;<br/><a NAME="E16:63_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#K2" title="LTLAXIO1:func.2">'X'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">Q</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"><a href="ltlaxio1.html#K2" title="LTLAXIO1:func.2">'X'</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:63_1"><span class="lab"><font color="Green" title="E38">A11</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="E17:63_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="c6">R</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#K2" title="LTLAXIO1:func.2">'X'</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#T47" onmouseover="rs('ltlaxio1/T47')" onmouseout="rh()">LTLAXIO1:47</a>, <a class="txt" href="#E8:63_1"><span class="lab"><font color="Green" title="E36">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>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> 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="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K2" title="LTLAXIO1:func.2">'X'</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>)</span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> 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="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K2" title="LTLAXIO1:func.2">'X'</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>)</span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:63_2"/>
<font color="Maroon" title="c6">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K2" title="LTLAXIO1:func.2">'X'</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>)</span>)</span></span><br/><a NAME="E2:63_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a> 
 <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="#E1:63"><span class="lab"><font color="Green" title="E32">A1</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">R</font> being   <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</a><span class="kw"> such that </span><br/><a NAME="E4:63_2"/><span class="lab"><font color="Green" title="E34">A12</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">R</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a> 
 <span class="kw">and </span><br/><a NAME="E5:63_2"/><span class="lab"><font color="Green" title="E35">A13</font></span>: 
<font color="Maroon" title="c7">R</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Maroon" title="c4">T</font>
 ;<br/><a NAME="E6:63_2"/>
 ex <font color="Olive" title="b1">t</font> being   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c4">T</font> st <br/>( <font color="Maroon" title="c7">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b1">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:63_2"><span class="lab"><font color="Green" title="E35">A13</font></span></a></span>;<br/><a NAME="E7:63_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">R</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/><span class="kw">hence </span><a NAME="E8:63_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="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K2" title="LTLAXIO1:func.2">'X'</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E4:63_2"><span class="lab"><font color="Green" title="E34">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>
<span class="kw">hence </span><a NAME="E4:63"/>
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>) <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K2" title="LTLAXIO1:func.2">'X'</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T57" onmouseover="rs('ltlaxio2/T57')" onmouseout="rh()">LTLAXIO2:57</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>
