let dA1, dA2 be ManySortedSet of Positives A; :: thesis: ( ex S being c=-monotone Function-yielding Sequence st
( dom S = succ A & dA1 = S . A & ( for B being Ordinal st B in succ A holds
ex SB being ManySortedSet of Positives B st
( S . B = SB & ( for x being object st x in Positives B holds
SB . x = [(Union (divL (||.x.||,(union (rng (S | B)))))),(Union (divR (||.x.||,(union (rng (S | B))))))] ) ) ) ) & ex S being c=-monotone Function-yielding Sequence st
( dom S = succ A & dA2 = S . A & ( for B being Ordinal st B in succ A holds
ex SB being ManySortedSet of Positives B st
( S . B = SB & ( for x being object st x in Positives B holds
SB . x = [(Union (divL (||.x.||,(union (rng (S | B)))))),(Union (divR (||.x.||,(union (rng (S | B))))))] ) ) ) ) implies dA1 = dA2 )
deffunc H₁( Ordinal) -> Subset of (Day $1) = Positives $1;
deffunc H₂( object , c=-monotone Function-yielding Sequence) -> object = [(Union (divL (||.$1.||,(union (rng $2))))),(Union (divR (||.$1.||,(union (rng $2)))))];
given S1 being c=-monotone Function-yielding Sequence such that A31: ( dom S1 = succ A & dA1 = S1 . A & ( for B being Ordinal st B in succ A holds
ex SB being ManySortedSet of Positives B st
( S1 . B = SB & ( for x being object st x in Positives B holds
SB . x = H₂(x,S1 | B) ) ) ) ) ; :: thesis: ( for S being c=-monotone Function-yielding Sequence holds
( not dom S = succ A or not dA2 = S . A or ex B being Ordinal st
( B in succ A & ( for SB being ManySortedSet of Positives B holds
( not S . B = SB or ex x being object st
( x in Positives B & not SB . x = [(Union (divL (||.x.||,(union (rng (S | B)))))),(Union (divR (||.x.||,(union (rng (S | B))))))] ) ) ) ) ) or dA1 = dA2 )
given S2 being c=-monotone Function-yielding Sequence such that A32: ( dom S2 = succ A & dA2 = S2 . A & ( for B being Ordinal st B in succ A holds
ex SB being ManySortedSet of Positives B st
( S2 . B = SB & ( for x being object st x in Positives B holds
SB . x = H₂(x,S2 | B) ) ) ) ) ; :: thesis: dA1 = dA2
A33: ( succ A c= dom S1 & succ A c= dom S2 ) by A31, A32;
A34: for B being Ordinal st B in succ A holds
ex SB being ManySortedSet of H₁(B) st
( S1 . B = SB & ( for o being object st o in H₁(B) holds
SB . o = H₂(o,S1 | B) ) ) by A31;
A35: for B being Ordinal st B in succ A holds
ex SB being ManySortedSet of H₁(B) st
( S2 . B = SB & ( for o being object st o in H₁(B) holds
SB . o = H₂(o,S2 | B) ) ) by A32;
S1 | (succ A) = S2 | (succ A) from SURREALR:sch 2(A33, A34, A35);
then S1 | (succ A) = S2 by A32;
hence dA1 = dA2 by A31, A32; :: thesis: verum