let A, B be Element of LTLB_WFF ; :: thesis: for F being Subset of LTLB_WFF st F |=0 'G' A & F |=0 'G' (A => B) holds
F |=0 'G' B
let F be Subset of LTLB_WFF; :: thesis: ( F |=0 'G' A & F |=0 'G' (A => B) implies F |=0 'G' B )
assume that
A1: F |=0 'G' A and
A2: F |=0 'G' (A => B) ; :: thesis: F |=0 'G' B
let M be LTLModel; :: according to LTLAXIO5:def 3 :: thesis: ( M |=0 F implies M |=0 'G' B )
assume B10: M |=0 F ; :: thesis: M |=0 'G' B
then B11: M |=0 'G' A by A1;
B12: M |=0 'G' (A => B) by B10, A2;

now :: thesis: for i being Element of NAT holds (SAT M) . [(0 + i),B] = 1

let i be Element of NAT ; :: thesis: (SAT M) . [(0 + i),B] = 1
B13: (SAT M) . [(0 + i),A] = 1 by B11, LTLAXIO1:10;
(SAT M) . [(0 + i),(A => B)] = 1 by B12, LTLAXIO1:10;
then ((SAT M) . [i,A]) => ((SAT M) . [i,B]) = 1 by LTLAXIO1:def 11;
hence (SAT M) . [(0 + i),B] = 1 by B13; :: thesis: verum

end;

hence M |=0 'G' B by LTLAXIO1:10; :: thesis: verum