deffunc H₁( set , Sequence) -> set = union (rng $2);
deffunc H₂( Ordinal, set ) -> set = new_set2 $2;
let A be non empty set ; :: thesis: for O being Ordinal
for T being Sequence st O <> 0 & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveSet2 (A,O1) ) holds
ConsecutiveSet2 (A,O) = union (rng T)
let O be Ordinal; :: thesis: for T being Sequence st O <> 0 & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveSet2 (A,O1) ) holds
ConsecutiveSet2 (A,O) = union (rng T)
let T be Sequence; :: thesis: ( O <> 0 & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveSet2 (A,O1) ) implies ConsecutiveSet2 (A,O) = union (rng T) )
deffunc H₃( Ordinal) -> set = ConsecutiveSet2 (A,$1);
assume that
A1: ( O <> 0 & O is limit_ordinal ) and
A2: dom T = O and
A3: for O1 being Ordinal st O1 in O holds
T . O1 = H₃(O1) ; :: thesis: ConsecutiveSet2 (A,O) = union (rng T)
A4: for O being Ordinal
for It being object holds
( It = H₃(O) iff ex L0 being Sequence st
( It = last L0 & dom L0 = succ O & L0 . 0 = A & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = H₂(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> 0 & C is limit_ordinal holds
L0 . C = H₁(C,L0 | C) ) ) ) by Def5;
thus H₃(O) = H₁(O,T) from ORDINAL2:sch 10(A4, A1, A2, A3); :: thesis: verum