deffunc H₁( Ordinal, Sequence) -> Element of the carrier of L = "/\" ((rng $2),L);
deffunc H₂( Ordinal, set ) -> set = f . $2;
( ex z being object ex S being Sequence st
( z = last S & dom S = succ O & S . 0 = x & ( for C being Ordinal st succ C in succ O holds
S . (succ C) = H₂(C,S . C) ) & ( for C being Ordinal st C in succ O & C <> 0 & C is limit_ordinal holds
S . C = H₁(C,S | C) ) ) & ( for x1, x2 being set st ex L0 being Sequence st
( x1 = last L0 & dom L0 = succ O & L0 . 0 = x & ( 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) ) ) & ex L0 being Sequence st
( x2 = last L0 & dom L0 = succ O & L0 . 0 = x & ( 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) ) ) holds
x1 = x2 ) ) from ORDINAL2:sch 7();
hence ( ex b₁ being set ex L0 being Sequence st
( b₁ = last L0 & dom L0 = succ O & L0 . 0 = x & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = f . (L0 . C) ) & ( for C being Ordinal st C in succ O & C <> 0 & C is limit_ordinal holds
L0 . C = "/\" ((rng (L0 | C)),L) ) ) & ( for b₁, b₂ being set st ex L0 being Sequence st
( b₁ = last L0 & dom L0 = succ O & L0 . 0 = x & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = f . (L0 . C) ) & ( for C being Ordinal st C in succ O & C <> 0 & C is limit_ordinal holds
L0 . C = "/\" ((rng (L0 | C)),L) ) ) & ex L0 being Sequence st
( b₂ = last L0 & dom L0 = succ O & L0 . 0 = x & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = f . (L0 . C) ) & ( for C being Ordinal st C in succ O & C <> 0 & C is limit_ordinal holds
L0 . C = "/\" ((rng (L0 | C)),L) ) ) holds
b₁ = b₂ ) ) ; :: thesis: verum