let L be Lattice; :: thesis: for f being Function of the carrier of L,the carrier of L
for x being Element of L
for O being Ordinal holds f,(succ O) +. x = f . (f,O +. x)
let f be Function of the carrier of L,the carrier of L; :: thesis: for x being Element of L
for O being Ordinal holds f,(succ O) +. x = f . (f,O +. x)
let x be Element of L; :: thesis: for O being Ordinal holds f,(succ O) +. x = f . (f,O +. x)
let O be Ordinal; :: thesis: f,(succ O) +. x = f . (f,O +. x)
deffunc H₁( Ordinal, set ) -> set = f . $2;
deffunc H₂( Ordinal, T-Sequence) -> Element of the carrier of L = "\/" (rng $2),L;
deffunc H₃( Ordinal) -> set = f,$1 +. x;
A1: for O being Ordinal
for y being set holds
( y = H₃(O) iff ex L0 being T-Sequence st
( y = last L0 & dom L0 = succ O & L0 . {} = x & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = H₁(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = H₂(C,L0 | C) ) ) ) by Def6;
for O being Ordinal holds H₃( succ O) = H₁(O,H₃(O)) from ORDINAL2:sch 9(A1);
hence f,(succ O) +. x = f . (f,O +. x) ; :: thesis: verum