let A be Element of LTLB_WFF ; :: thesis: for F being Subset of LTLB_WFF holds
( F |= A iff 'G' F |=0 A )
let F be Subset of LTLB_WFF; :: thesis: ( F |= A iff 'G' F |=0 A )

hereby :: thesis: ( 'G' F |=0 A implies F |= A )

assume Z1: F |= A ; :: thesis: 'G' F |=0 A
thus 'G' F |=0 A :: thesis: verum

proof

let M be LTLModel; :: according to LTLAXIO5:def 3 :: thesis: ( M |=0 'G' F implies M |=0 A )
assume M |=0 'G' F ; :: thesis: M |=0 A
then M |= A by Z1, th261bq;
hence M |=0 A ; :: thesis: verum

end;

end;

assume Z2: 'G' F |=0 A ; :: thesis: F |= A
thus F |= A :: thesis: verum

proof

let M be LTLModel; :: according to LTLAXIO1:def 14 :: thesis: ( not M |= F or M |= A )
assume Z3: M |= F ; :: thesis: M |= A
let i be Element of NAT ; :: according to LTLAXIO1:def 12 :: thesis: (SAT M) . [i,A] = 1
M ^\ i |= F by LTLAXIO1:29, Z3;
then M ^\ i |=0 A by Z2, th261bq;
then (SAT M) . [(i + 0),A] = 1 by LTLAXIO1:28;
hence (SAT M) . [i,A] = 1 ; :: thesis: verum

end;