deffunc H₁( set , T-Sequence) -> set = union (rng $2);
let A be non empty set ; :: thesis: for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being QuadrSeq of d
for T being T-Sequence
for O being Ordinal st O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta2 q,O1 ) holds
ConsecutiveDelta2 q,O = union (rng T)
let L be lower-bounded LATTICE; :: thesis: for d being BiFunction of A,L
for q being QuadrSeq of d
for T being T-Sequence
for O being Ordinal st O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta2 q,O1 ) holds
ConsecutiveDelta2 q,O = union (rng T)
let d be BiFunction of A,L; :: thesis: for q being QuadrSeq of d
for T being T-Sequence
for O being Ordinal st O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta2 q,O1 ) holds
ConsecutiveDelta2 q,O = union (rng T)
let q be QuadrSeq of d; :: thesis: for T being T-Sequence
for O being Ordinal st O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta2 q,O1 ) holds
ConsecutiveDelta2 q,O = union (rng T)
let T be T-Sequence; :: thesis: for O being Ordinal st O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta2 q,O1 ) holds
ConsecutiveDelta2 q,O = union (rng T)
let O be Ordinal; :: thesis: ( O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta2 q,O1 ) implies ConsecutiveDelta2 q,O = union (rng T) )
deffunc H₂( Ordinal, set ) -> BiFunction of (new_set2 (ConsecutiveSet2 A,$1)),L = new_bi_fun2 (BiFun $2,(ConsecutiveSet2 A,$1),L),(Quadr2 q,$1);
deffunc H₃( Ordinal) -> set = ConsecutiveDelta2 q,$1;
assume that
A1: ( O <> {} & 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: ConsecutiveDelta2 q,O = union (rng T)
A4: for O being Ordinal
for It being set holds
( It = H₃(O) iff ex L0 being T-Sequence st
( It = last L0 & dom L0 = succ O & L0 . {} = 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 <> {} & C is limit_ordinal holds
L0 . C = H₁(C,L0 | C) ) ) ) by Def8;
thus H₃(O) = H₁(O,T) from ORDINAL2:sch 10(A4, A1, A2, A3); :: thesis: verum