deffunc H₁( Ordinal) -> Subset of [:(Day $1),(Day $1):] = Triangle $1;
deffunc H₂( object , Function-yielding c=-monotone Sequence) -> object = [((union (rng $2)) .: ([:(L_ (L_ $1)),{(R_ $1)}:] \/ [:{(L_ $1)},(L_ (R_ $1)):])),((union (rng $2)) .: ([:(R_ (L_ $1)),{(R_ $1)}:] \/ [:{(L_ $1)},(R_ (R_ $1)):]))];
let it1, it2 be ManySortedSet of Triangle A; :: thesis: ( ex S being Function-yielding c=-monotone Sequence st
( dom S = succ A & it1 = S . A & ( for B being Ordinal st B in succ A holds
ex SB being ManySortedSet of Triangle B st
( S . B = SB & ( for x being object st x in Triangle B holds
SB . x = [((union (rng (S | B))) .: ([:(L_ (L_ x)),{(R_ x)}:] \/ [:{(L_ x)},(L_ (R_ x)):])),((union (rng (S | B))) .: ([:(R_ (L_ x)),{(R_ x)}:] \/ [:{(L_ x)},(R_ (R_ x)):]))] ) ) ) ) & ex S being Function-yielding c=-monotone Sequence st
( dom S = succ A & it2 = S . A & ( for B being Ordinal st B in succ A holds
ex SB being ManySortedSet of Triangle B st
( S . B = SB & ( for x being object st x in Triangle B holds
SB . x = [((union (rng (S | B))) .: ([:(L_ (L_ x)),{(R_ x)}:] \/ [:{(L_ x)},(L_ (R_ x)):])),((union (rng (S | B))) .: ([:(R_ (L_ x)),{(R_ x)}:] \/ [:{(L_ x)},(R_ (R_ x)):]))] ) ) ) ) implies it1 = it2 )
given S1 being Function-yielding c=-monotone Sequence such that A27: ( dom S1 = succ A & it1 = S1 . A ) and
A28: for B being Ordinal st B in succ A holds
ex SB being ManySortedSet of H₁(B) st
( S1 . B = SB & ( for x being object st x in H₁(B) holds
SB . x = H₂(x,S1 | B) ) ) ; :: thesis: ( for S being Function-yielding c=-monotone Sequence holds
( not dom S = succ A or not it2 = S . A or ex B being Ordinal st
( B in succ A & ( for SB being ManySortedSet of Triangle B holds
( not S . B = SB or ex x being object st
( x in Triangle B & not SB . x = [((union (rng (S | B))) .: ([:(L_ (L_ x)),{(R_ x)}:] \/ [:{(L_ x)},(L_ (R_ x)):])),((union (rng (S | B))) .: ([:(R_ (L_ x)),{(R_ x)}:] \/ [:{(L_ x)},(R_ (R_ x)):]))] ) ) ) ) ) or it1 = it2 )
given S2 being Function-yielding c=-monotone Sequence such that A29: ( dom S2 = succ A & it2 = S2 . A ) and
A30: for B being Ordinal st B in succ A holds
ex SB being ManySortedSet of H₁(B) st
( S2 . B = SB & ( for x being object st x in H₁(B) holds
SB . x = H₂(x,S2 | B) ) ) ; :: thesis: it1 = it2
A31: ( succ A c= dom S1 & succ A c= dom S2 ) by A27, A29;
S1 | (succ A) = S2 | (succ A) from SURREALR:sch 2(A31, A28, A30);
then S1 | (succ A) = S2 by A29;
hence it1 = it2 by A27, A29; :: thesis: verum