let A, B be Element of LTLB_WFF ; :: thesis: for F being Subset of LTLB_WFF st F |= A => B & F |= A => ('X' A) holds
F |= A => ('G' B)
let F be Subset of LTLB_WFF; :: thesis: ( F |= A => B & F |= A => ('X' A) implies F |= A => ('G' B) )
assume that
A1: F |= A => B and
A2: F |= A => ('X' A) ; :: thesis: F |= A => ('G' B)
let M be LTLModel; :: according to LTLAXIO1:def 14 :: thesis: ( M |= F implies M |= A => ('G' B) )
assume A3: M |= F ; :: thesis: M |= A => ('G' B)
let n be Element of NAT ; :: according to LTLAXIO1:def 12 :: thesis: (SAT M) . [n,(A => ('G' B))] = 1
defpred S₁[ Nat] means (SAT M) . [(n + $1),A] = 1;

per cases ( (SAT M) . [n,A] = 1 or (SAT M) . [n,A] = 0 ) by XBOOLEAN:def 3;

suppose A4: (SAT M) . [n,A] = 1 ; :: thesis: (SAT M) . [n,(A => ('G' B))] = 1

A5: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A6: S₁[k] ; :: thesis: S₁[k + 1]
reconsider kk = k as Element of NAT by ORDINAL1:def 12;
(SAT M) . [(n + kk),(A => ('X' A))] = 1 by A2, A3, Def12;
then ((SAT M) . [(n + kk),A]) => ((SAT M) . [(n + kk),('X' A)]) = 1 by Def11;
then (SAT M) . [((n + kk) + 1),A] = 1 by A6, Th9;
hence S₁[k + 1] ; :: thesis: verum

end;

A7: S₁[ 0 ] by A4;
A8: for i being Nat holds S₁[i] from NAT_1:sch 2(A7, A5);

now :: thesis: for i being Element of NAT holds (SAT M) . [(n + i),B] = 1

let i be Element of NAT ; :: thesis: (SAT M) . [(n + i),B] = 1
(SAT M) . [(n + i),(A => B)] = 1 by A3, A1, Def12;
then A9: ((SAT M) . [(n + i),A]) => ((SAT M) . [(n + i),B]) = 1 by Def11;
(SAT M) . [(n + i),A] = 1 by A8;
hence (SAT M) . [(n + i),B] = 1 by A9; :: thesis: verum

end;

then (SAT M) . [n,('G' B)] = 1 by Th10;
then ((SAT M) . [n,A]) => ((SAT M) . [n,('G' B)]) = 1 ;
hence (SAT M) . [n,(A => ('G' B))] = 1 by Def11; :: thesis: verum

end;

suppose (SAT M) . [n,A] = 0 ; :: thesis: (SAT M) . [n,(A => ('G' B))] = 1

then ((SAT M) . [n,A]) => ((SAT M) . [n,('G' B)]) = 1 ;
hence (SAT M) . [n,(A => ('G' B))] = 1 by Def11; :: thesis: verum

end;

end;