let L be Lattice; 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 Sequence st O <> 0 & 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; for x being Element of L
for O being Ordinal
for T being Sequence st O <> 0 & 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; for O being Ordinal
for T being Sequence st O <> 0 & 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; for T being Sequence st O <> 0 & 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 Sequence; ( O <> 0 & 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, Sequence) -> Element of the carrier of L = "\/" ((rng $2),L);
deffunc H3( Ordinal) -> set = (f,$1) +. x;
assume that
A1:
( O <> 0 & O is limit_ordinal )
and
A2:
dom T = O
and
A3:
for A being Ordinal st A in O holds
T . A = H3(A)
; (f,O) +. x = "\/" ((rng T),L)
A4:
for O being Ordinal
for y being object holds
( y = H3(O) iff ex L0 being Sequence st
( y = last L0 & dom L0 = succ O & L0 . 0 = 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 <> 0 & C is limit_ordinal holds
L0 . C = H2(C,L0 | C) ) ) )
by Def4;
thus
H3(O) = H2(O,T)
from ORDINAL2:sch 10(A4, A1, A2, A3); verum