let v be LTL-formula; :: thesis: for w being Element of Inf_seq AtomicFamily
for f being Function of (LTLNodes v),(LTLNodes v) st f is_succ_homomorphism v,w holds
for x being set st x in LTLNodes v & not CastNode x,v is elementary & w |= * (CastNode x,v) holds
for k being Nat st ( for i being Nat st i <= k holds
not CastNode ((f |** i) . x),v is elementary ) holds
( CastNode ((f |** (k + 1)) . x),v is_succ_of CastNode ((f |** k) . x),v & w |= * (CastNode ((f |** k) . x),v) )
let w be Element of Inf_seq AtomicFamily ; :: thesis: for f being Function of (LTLNodes v),(LTLNodes v) st f is_succ_homomorphism v,w holds
for x being set st x in LTLNodes v & not CastNode x,v is elementary & w |= * (CastNode x,v) holds
for k being Nat st ( for i being Nat st i <= k holds
not CastNode ((f |** i) . x),v is elementary ) holds
( CastNode ((f |** (k + 1)) . x),v is_succ_of CastNode ((f |** k) . x),v & w |= * (CastNode ((f |** k) . x),v) )
set LN = LTLNodes v;
let f be Function of (LTLNodes v),(LTLNodes v); :: thesis: ( f is_succ_homomorphism v,w implies for x being set st x in LTLNodes v & not CastNode x,v is elementary & w |= * (CastNode x,v) holds
for k being Nat st ( for i being Nat st i <= k holds
not CastNode ((f |** i) . x),v is elementary ) holds
( CastNode ((f |** (k + 1)) . x),v is_succ_of CastNode ((f |** k) . x),v & w |= * (CastNode ((f |** k) . x),v) ) )
assume A1: f is_succ_homomorphism v,w ; :: thesis: for x being set st x in LTLNodes v & not CastNode x,v is elementary & w |= * (CastNode x,v) holds
for k being Nat st ( for i being Nat st i <= k holds
not CastNode ((f |** i) . x),v is elementary ) holds
( CastNode ((f |** (k + 1)) . x),v is_succ_of CastNode ((f |** k) . x),v & w |= * (CastNode ((f |** k) . x),v) )
then A2: f is_homomorphism v,w by Th46;
for x being set st x in LTLNodes v & not CastNode x,v is elementary & w |= * (CastNode x,v) holds
for k being Nat st ( for i being Nat st i <= k holds
not CastNode ((f |** i) . x),v is elementary ) holds
( CastNode ((f |** (k + 1)) . x),v is_succ_of CastNode ((f |** k) . x),v & w |= * (CastNode ((f |** k) . x),v) )

proof

let x be set ; :: thesis: ( x in LTLNodes v & not CastNode x,v is elementary & w |= * (CastNode x,v) implies for k being Nat st ( for i being Nat st i <= k holds
not CastNode ((f |** i) . x),v is elementary ) holds
( CastNode ((f |** (k + 1)) . x),v is_succ_of CastNode ((f |** k) . x),v & w |= * (CastNode ((f |** k) . x),v) ) )
assume that
A3: x in LTLNodes v and
A4: ( not CastNode x,v is elementary & w |= * (CastNode x,v) ) ; :: thesis: for k being Nat st ( for i being Nat st i <= k holds
not CastNode ((f |** i) . x),v is elementary ) holds
( CastNode ((f |** (k + 1)) . x),v is_succ_of CastNode ((f |** k) . x),v & w |= * (CastNode ((f |** k) . x),v) )
for k being Nat st ( for i being Nat st i <= k holds
not CastNode ((f |** i) . x),v is elementary ) holds
( CastNode ((f |** (k + 1)) . x),v is_succ_of CastNode ((f |** k) . x),v & w |= * (CastNode ((f |** k) . x),v) )

proof

let k be Nat; :: thesis: ( ( for i being Nat st i <= k holds
not CastNode ((f |** i) . x),v is elementary ) implies ( CastNode ((f |** (k + 1)) . x),v is_succ_of CastNode ((f |** k) . x),v & w |= * (CastNode ((f |** k) . x),v) ) )
assume A5: for i being Nat st i <= k holds
not CastNode ((f |** i) . x),v is elementary ; :: thesis: ( CastNode ((f |** (k + 1)) . x),v is_succ_of CastNode ((f |** k) . x),v & w |= * (CastNode ((f |** k) . x),v) )
set y = (f |** k) . x;
A6: (f |** k) . x in LTLNodes v by A3, FUNCT_2:7;
A7: (f |** (k + 1)) . x = (f * (f |** k)) . x by FUNCT_7:73
.= f . ((f |** k) . x) by A3, FUNCT_2:21 ;
( not CastNode ((f |** k) . x),v is elementary & w |= * (CastNode ((f |** k) . x),v) ) by A2, A3, A4, A5, Th47;
hence ( CastNode ((f |** (k + 1)) . x),v is_succ_of CastNode ((f |** k) . x),v & w |= * (CastNode ((f |** k) . x),v) ) by A1, A6, A7, Def32; :: thesis: verum

end;

hence for k being Nat st ( for i being Nat st i <= k holds
not CastNode ((f |** i) . x),v is elementary ) holds
( CastNode ((f |** (k + 1)) . x),v is_succ_of CastNode ((f |** k) . x),v & w |= * (CastNode ((f |** k) . x),v) ) ; :: thesis: verum

end;

hence for x being set st x in LTLNodes v & not CastNode x,v is elementary & w |= * (CastNode x,v) holds
for k being Nat st ( for i being Nat st i <= k holds
not CastNode ((f |** i) . x),v is elementary ) holds
( CastNode ((f |** (k + 1)) . x),v is_succ_of CastNode ((f |** k) . x),v & w |= * (CastNode ((f |** k) . x),v) ) ; :: thesis: verum