let L be Lattice; for f being Function of the carrier of L, the carrier of L
for x being Element of L holds (f,0) -. x = x
let f be Function of the carrier of L, the carrier of L; for x being Element of L holds (f,0) -. x = x
let x be Element of L; (f,0) -. x = x
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;
A1:
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 Def5;
thus
H3( 0 ) = x
from ORDINAL2:sch 8(A1); verum