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
for T being T-Sequence st O <> {} & O is limit_ordinal & dom T = O & ( for A being Ordinal st A in O holds
T . A = f,A +. x ) holds
f,O +. x = "\/" (rng T),L
let f be Function of the carrier of L,the carrier of L; :: thesis: for x being Element of L
for O being Ordinal
for T being T-Sequence st O <> {} & O is limit_ordinal & dom T = O & ( for A being Ordinal st A in O holds
T . A = f,A +. x ) holds
f,O +. x = "\/" (rng T),L
let x be Element of L; :: thesis: for O being Ordinal
for T being T-Sequence st O <> {} & O is limit_ordinal & dom T = O & ( for A being Ordinal st A in O holds
T . A = f,A +. x ) holds
f,O +. x = "\/" (rng T),L
let O be Ordinal; :: thesis: for T being T-Sequence st O <> {} & O is limit_ordinal & dom T = O & ( for A being Ordinal st A in O holds
T . A = f,A +. x ) holds
f,O +. x = "\/" (rng T),L
let T be T-Sequence; :: thesis: ( O <> {} & O is limit_ordinal & dom T = O & ( for A being Ordinal st A in O holds
T . A = f,A +. x ) implies f,O +. x = "\/" (rng T),L )
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;
assume that
A1: ( O <> {} & O is limit_ordinal ) and
A2: dom T = O and
A3: for A being Ordinal st A in O holds
T . A = H₃(A) ; :: thesis: f,O +. x = "\/" (rng T),L
A4: 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;
thus H₃(O) = H₂(O,T) from ORDINAL2:sch 10(A4, A1, A2, A3); :: thesis: verum