let A be non empty set ; :: thesis: for L being lower-bounded LATTICE
for T being Sequence
for O being Ordinal
for d being BiFunction of A,L
for q being QuadrSeq of d st O <> 0 & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta (q,O1) ) holds
ConsecutiveDelta (q,O) = union (rng T)
let L be lower-bounded LATTICE; :: thesis: for T being Sequence
for O being Ordinal
for d being BiFunction of A,L
for q being QuadrSeq of d st O <> 0 & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta (q,O1) ) holds
ConsecutiveDelta (q,O) = union (rng T)
let T be Sequence; :: thesis: for O being Ordinal
for d being BiFunction of A,L
for q being QuadrSeq of d st O <> 0 & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta (q,O1) ) holds
ConsecutiveDelta (q,O) = union (rng T)
let O be Ordinal; :: thesis: for d being BiFunction of A,L
for q being QuadrSeq of d st O <> 0 & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta (q,O1) ) holds
ConsecutiveDelta (q,O) = union (rng T)
deffunc H₁( Ordinal, Sequence) -> set = union (rng $2);
let d be BiFunction of A,L; :: thesis: for q being QuadrSeq of d st O <> 0 & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta (q,O1) ) holds
ConsecutiveDelta (q,O) = union (rng T)
let q be QuadrSeq of d; :: thesis: ( O <> 0 & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta (q,O1) ) implies ConsecutiveDelta (q,O) = union (rng T) )
deffunc H₂( Ordinal, set ) -> BiFunction of (new_set (ConsecutiveSet (A,$1))),L = new_bi_fun ((BiFun ($2,(ConsecutiveSet (A,$1)),L)),(Quadr (q,$1)));
deffunc H₃( Ordinal) -> set = ConsecutiveDelta (q,$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: ConsecutiveDelta (q,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 = d & ( 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 Def16;
thus H₃(O) = H₁(O,T) from ORDINAL2:sch 10(A4, A1, A2, A3); :: thesis: verum