<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">A</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" 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="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>  st <font color="Olive" title="b1">F</font> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Olive" title="b1">F</font> <a href="ltlaxio1.html#R3" title="LTLAXIO1:pred.3">|=</a> <font color="Maroon" title="c1">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon" title="c1">A</font> implies <font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R3" title="LTLAXIO1:pred.3">|=</a> <font color="Maroon" title="c1">A</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:69"/>
<font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R8" title="LTLAXIO1:pred.8">|-</a> <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R3" title="LTLAXIO1:pred.3">|=</a> <font color="Maroon" title="c1">A</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">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> <span class="kw"> such that </span><br/><a NAME="E3:69"/><span class="lab"><font color="Green" title="E59">A1</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E4:69"/><span class="lab"><font color="Green" title="E60">A2</font></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> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E5:69"/><span class="lab"><font color="Green" title="E61">A3</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> holds <br/> <a href="ltlaxio1.html#R7" title="LTLAXIO1:pred.7">prc</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">F</font>,<font color="Olive" title="b1">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D30" target="_self" title="LTLAXIO1:def.30">Def31</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; $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="c3">f</font> implies <font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R3" title="LTLAXIO1:pred.3">|=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 );<br/>
<a NAME="E6:69"/><span class="lab"><font color="Green" title="E62">A4</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st (  for <font color="Olive" title="b2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">j</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" 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="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:69_1"/><span class="lab"><font color="Green" title="E63">A5</font></span>: 
 for <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font>]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:69_1_1"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  not <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:69_1_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font>]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:69_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font>]
 ; <a class="txt" 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="69_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:69_1_1_2"/>
 not <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font>]</span><br/><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:69_1_1_2"/><span class="lab"><font color="Green" title="E64">A6</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">and </span><br/><a NAME="E3:69_1_1_2"/><span class="lab"><font color="Green" title="E65">A7</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R3" title="LTLAXIO1:pred.3">|=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span><br/><a NAME="E4:69_1_1_2"/><span class="lab"><font color="Green" title="E66">A8</font></span>: 
 <a href="ltlaxio1.html#R7" title="LTLAXIO1:pred.7">prc</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:69"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="txt" href="#E2:69_1_1_2"><span class="lab"><font color="Green" title="E64">A6</font></span></a>, <a class="txt" href="#E3:69_1_1_2"><span class="lab"><font color="Green" title="E65">A7</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:69_1_1_2_1"/>
( <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ltlaxio1.html#K13" title="LTLAXIO1:func.13">LTL_axioms</a>  or <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> or  ex <font color="Olive" title="b1">j</font>, <font color="Olive" title="b2">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> &amp; ( <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font>,<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">k</font> <a href="ltlaxio1.html#R5" title="LTLAXIO1:pred.5">MP_rule</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> or <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font>,<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">k</font> <a href="ltlaxio1.html#R6" title="LTLAXIO1:pred.6">IND_rule</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> ) ) or  ex <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font> <a href="ltlaxio1.html#R4" title="LTLAXIO1:pred.4">NEX_rule</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_1_1_2"><span class="lab"><font color="Green" title="E66">A8</font></span></a>, <a class="ref" href="ltlaxio1.html#D29" target="_self" title="LTLAXIO1:def.29">Def30</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:69_1_1_2_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ltlaxio1.html#K13" title="LTLAXIO1:func.13">LTL_axioms</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R3" title="LTLAXIO1:pred.3">|=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span><br/><div class="add"><a NAME="E2:69_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A9</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN: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="#E2:69_1_1_2"><span class="lab"><font color="Green" title="E64">A6</font></span></a>, <a class="txt" href="#E3:69_1_1_2"><span class="lab"><font color="Green" title="E65">A7</font></span></a>, <a class="txt" href="ltlaxio1.html#E1"><span class="lab"><font color="Green" title="E1">Th1</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:69_1_1_2_1_1_1"/>
( <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> is  <a href="ltlaxio1.html#V1" title="LTLAXIO1:attr.1">LTL_TAUT_OF_PL</a>  or <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> is  <a href="ltlaxio1.html#V3" title="LTLAXIO1:attr.3">axltl1</a>  or <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> is  <a href="ltlaxio1.html#V4" title="LTLAXIO1:attr.4">axltl1a</a>  or <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> is  <a href="ltlaxio1.html#V5" title="LTLAXIO1:attr.5">axltl2</a>  or <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> is  <a href="ltlaxio1.html#V6" title="LTLAXIO1:attr.6">axltl3</a>  or <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> is  <a href="ltlaxio1.html#V7" title="LTLAXIO1:attr.7">axltl4</a>  or <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> is  <a href="ltlaxio1.html#V8" title="LTLAXIO1:attr.8">axltl5</a>  or <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> is  <a href="ltlaxio1.html#V9" title="LTLAXIO1:attr.9">axltl6</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69_1_1_2_1_1"><span class="lab"><font color="Green" title="E67">A9</font></span></a>, <a class="ref" href="ltlaxio1.html#T36" target="_self" title="LTLAXIO1:th.36">Th38</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:69_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> is  <a href="ltlaxio1.html#V1" title="LTLAXIO1:attr.1">LTL_TAUT_OF_PL</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R3" title="LTLAXIO1:pred.3">|=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:69_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R3" title="LTLAXIO1:pred.3">|=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#T33" target="_self" title="LTLAXIO1:th.33">Th34</a></span>; <a class="txt" 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="69_1_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:69_1_1_2_1_1_1_2"/>
( <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> is  <a href="ltlaxio1.html#V3" title="LTLAXIO1:attr.3">axltl1</a>  or <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> is  <a href="ltlaxio1.html#V4" title="LTLAXIO1:attr.4">axltl1a</a>  or <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> is  <a href="ltlaxio1.html#V5" title="LTLAXIO1:attr.5">axltl2</a>  or <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> is  <a href="ltlaxio1.html#V6" title="LTLAXIO1:attr.6">axltl3</a>  or <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> is  <a href="ltlaxio1.html#V7" title="LTLAXIO1:attr.7">axltl4</a>  or <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> is  <a href="ltlaxio1.html#V8" title="LTLAXIO1:attr.8">axltl5</a>  or <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> is  <a href="ltlaxio1.html#V9" title="LTLAXIO1:attr.9">axltl6</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R3" title="LTLAXIO1:pred.3">|=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:69_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R3" title="LTLAXIO1:pred.3">|=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#T37" target="_self" title="LTLAXIO1:th.37">Th39</a></span>; <a class="txt" 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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:69_1_1_2_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R3" title="LTLAXIO1:pred.3">|=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span><br/><div class="add"><a NAME="E2:69_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A10</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69_1_1_2"><span class="lab"><font color="Green" title="E64">A6</font></span></a>, <a class="txt" href="#E3:69_1_1_2"><span class="lab"><font color="Green" title="E65">A7</font></span></a>, <a class="txt" href="ltlaxio1.html#E1"><span class="lab"><font color="Green" title="E1">Th1</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E3:69_1_1_2_1_2"/>
<font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R3" title="LTLAXIO1:pred.3">|=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">M</font> be   <a href="ltlaxio1.html#NM1" title="LTLAXIO1:NM.1">LTLModel</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ltlaxio1.html#D14" target="_self" title="LTLAXIO1:def.14">LTLAXIO1:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">M</font> <a href="ltlaxio1.html#R2" title="LTLAXIO1:pred.2">|=</a> <font color="Maroon" title="c2">F</font> implies <font color="Maroon" title="c5">M</font> <a href="ltlaxio1.html#R1" title="LTLAXIO1:pred.1">|=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:69_1_1_2_1_2_1"/>
<font color="Maroon" title="c5">M</font> <a href="ltlaxio1.html#R2" title="LTLAXIO1:pred.2">|=</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">M</font> <a href="ltlaxio1.html#R1" title="LTLAXIO1:pred.1">|=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span><br/>

<span class="kw">hence </span><a NAME="E2:69_1_1_2_1_2_1"/>
<font color="Maroon" title="c5">M</font> <a href="ltlaxio1.html#R1" title="LTLAXIO1:pred.1">|=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69_1_1_2_1_2"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="ref" href="ltlaxio1.html#D13" target="_self" title="LTLAXIO1:def.13">Def14</a></span>; <a class="txt" 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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:69_1_1_2_1_3"/>
 ex <font color="Olive" title="b1">j</font>, <font color="Olive" title="b2">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> &amp; ( <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font>,<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">k</font> <a href="ltlaxio1.html#R5" title="LTLAXIO1:pred.5">MP_rule</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> or <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font>,<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">k</font> <a href="ltlaxio1.html#R6" title="LTLAXIO1:pred.6">IND_rule</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R3" title="LTLAXIO1:pred.3">|=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">j</font>, <font color="Maroon" title="c6">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:69_1_1_2_1_3"/><span class="lab"><font color="Green" title="E67">A11</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font>
 <span class="kw">and </span><br/><a NAME="E4:69_1_1_2_1_3"/><span class="lab"><font color="Green" title="E68">A12</font></span>: 
<font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">and </span><br/><a NAME="E5:69_1_1_2_1_3"/><span class="lab"><font color="Green" title="E69">A13</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">and </span><br/><a NAME="E6:69_1_1_2_1_3"/><span class="lab"><font color="Green" title="E70">A14</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">and </span><br/><a NAME="E7:69_1_1_2_1_3"/><span class="lab"><font color="Green" title="E71">A15</font></span>: 
( <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font> <a href="ltlaxio1.html#R5" title="LTLAXIO1:pred.5">MP_rule</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> or <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font> <a href="ltlaxio1.html#R6" title="LTLAXIO1:pred.6">IND_rule</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> )
 ;<br/><a NAME="E8:69_1_1_2_1_3"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1_1_2"><span class="lab"><font color="Green" title="E65">A7</font></span></a>, <a class="txt" href="#E6:69_1_1_2_1_3"><span class="lab"><font color="Green" title="E70">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:69_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A16</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R3" title="LTLAXIO1:pred.3">|=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="txt" href="#E5:69_1_1_2_1_3"><span class="lab"><font color="Green" title="E69">A13</font></span></a>, <a class="txt" href="#E6:69_1_1_2_1_3"><span class="lab"><font color="Green" title="E70">A14</font></span></a></span>;<br/><a NAME="E10:69_1_1_2_1_3"/><span class="lab"><font color="Green" title="E73">A17</font></span>: 
<font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1_1_2"><span class="lab"><font color="Green" title="E65">A7</font></span></a>, <a class="txt" href="#E4:69_1_1_2_1_3"><span class="lab"><font color="Green" title="E68">A12</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_2_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:69_1_1_2_1_3_1"/>
( <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font> <a href="ltlaxio1.html#R5" title="LTLAXIO1:pred.5">MP_rule</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> or <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font> <a href="ltlaxio1.html#R6" title="LTLAXIO1:pred.6">IND_rule</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:69_1_1_2_1_3"><span class="lab"><font color="Green" title="E71">A15</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_2_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:69_1_1_2_1_3_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font> <a href="ltlaxio1.html#R5" title="LTLAXIO1:pred.5">MP_rule</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R3" title="LTLAXIO1:pred.3">|=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span><br/><div class="add"><a NAME="E2:69_1_1_2_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R3" title="LTLAXIO1:pred.3">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:69_1_1_2_1_3"><span class="lab"><font color="Green" title="E72">A16</font></span></a>, <a class="ref" href="ltlaxio1.html#D20" target="_self" title="LTLAXIO1:def.20">Def21</a></span>;<br/><span class="kw">hence </span><a NAME="E3:69_1_1_2_1_3_1_1"/>
<font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R3" title="LTLAXIO1:pred.3">|=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="txt" href="#E3:69_1_1_2_1_3"><span class="lab"><font color="Green" title="E67">A11</font></span></a>, <a class="txt" href="#E4:69_1_1_2_1_3"><span class="lab"><font color="Green" title="E68">A12</font></span></a>, <a class="txt" href="#E10:69_1_1_2_1_3"><span class="lab"><font color="Green" title="E73">A17</font></span></a>, <a class="ref" href="ltlaxio1.html#T24" target="_self" title="LTLAXIO1:th.24">Th25</a></span>; <a class="txt" 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="69_1_1_2_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:69_1_1_2_1_3_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font> <a href="ltlaxio1.html#R6" title="LTLAXIO1:pred.6">IND_rule</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R3" title="LTLAXIO1:pred.3">|=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c8">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <span class="kw"> such that </span><br/><a NAME="E3:69_1_1_2_1_3_1_2"/><span class="lab"><font color="Green" title="E74">A18</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c8">B</font>
 <span class="kw">and </span><br/><a NAME="E4:69_1_1_2_1_3_1_2"/><span class="lab"><font color="Green" title="E75">A19</font></span>: 
( <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</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" title="c7">A</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K6" title="LTLAXIO1:func.6">'G'</a> <font color="Maroon" title="c8">B</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D21" target="_self" title="LTLAXIO1:def.21">Def22</a></span>;<br/><a NAME="E5:69_1_1_2_1_3_1_2"/>
<font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R3" title="LTLAXIO1:pred.3">|=</a> <font color="Maroon" title="c7">A</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c8">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="txt" href="#E3:69_1_1_2_1_3"><span class="lab"><font color="Green" title="E67">A11</font></span></a>, <a class="txt" href="#E4:69_1_1_2_1_3"><span class="lab"><font color="Green" title="E68">A12</font></span></a>, <a class="txt" href="#E10:69_1_1_2_1_3"><span class="lab"><font color="Green" title="E73">A17</font></span></a>, <a class="txt" href="#E3:69_1_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E74">A18</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:69_1_1_2_1_3_1_2"/>
<font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R3" title="LTLAXIO1:pred.3">|=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:69_1_1_2_1_3"><span class="lab"><font color="Green" title="E72">A16</font></span></a>, <a class="txt" href="#E4:69_1_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E75">A19</font></span></a>, <a class="ref" href="ltlaxio1.html#T27" target="_self" title="LTLAXIO1:th.27">Th28</a></span>; <a class="txt" 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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_2_1_4"><span class="kw">suppose </span></a><a NAME="E1:69_1_1_2_1_4"/>
 ex <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font> <a href="ltlaxio1.html#R4" title="LTLAXIO1:pred.4">NEX_rule</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R3" title="LTLAXIO1:pred.3">|=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:69_1_1_2_1_4"/><span class="lab"><font color="Green" title="E67">A20</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font>
 <span class="kw">and </span><br/><a NAME="E4:69_1_1_2_1_4"/><span class="lab"><font color="Green" title="E68">A21</font></span>: 
<font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">and </span><br/><a NAME="E5:69_1_1_2_1_4"/><span class="lab"><font color="Green" title="E69">A22</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font> <a href="ltlaxio1.html#R4" title="LTLAXIO1:pred.4">NEX_rule</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>
 ;<br/><a NAME="E6:69_1_1_2_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">j</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="txt" href="#E4:69_1_1_2_1_4"><span class="lab"><font color="Green" title="E68">A21</font></span></a></span>;<br/><a NAME="E7:69_1_1_2_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R3" title="LTLAXIO1:pred.3">|=</a>  <a href="ltlaxio1.html#K2" title="LTLAXIO1:func.2">'X'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1_1_2"><span class="lab"><font color="Green" title="E65">A7</font></span></a>, <a class="txt" href="#E3:69_1_1_2_1_4"><span class="lab"><font color="Green" title="E67">A20</font></span></a>, <a class="txt" href="#E4:69_1_1_2_1_4"><span class="lab"><font color="Green" title="E68">A21</font></span></a>, <a class="ref" href="ltlaxio1.html#T25" target="_self" title="LTLAXIO1:th.25">Th26</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E8:69_1_1_2_1_4"/>
<font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R3" title="LTLAXIO1:pred.3">|=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:69_1_1_2_1_4"><span class="lab"><font color="Green" title="E69">A22</font></span></a>, <a class="ref" href="ltlaxio1.html#D19" target="_self" title="LTLAXIO1:def.19">Def20</a></span>; <a class="txt" 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="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E7:69"/><span class="lab"><font color="Green" title="E63">A23</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S4" title="NAT_1:sch.4">NAT_1:sch 4</a>(<a class="txt" href="#E6:69"><span class="lab"><font color="Green" title="E62">A4</font></span></a>);<br/></span>
<a NAME="E8:69"/>
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="ltlaxio1.html#E1"><span class="lab"><font color="Green" title="E1">Th1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:69"/>
<font color="Maroon" title="c2">F</font> <a href="ltlaxio1.html#R3" title="LTLAXIO1:pred.3">|=</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E7:69"><span class="lab"><font color="Green" title="E63">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
