deffunc H_{1}( Ordinal, Sequence) -> Element of the carrier of L = "/\" ((rng $2),L);

deffunc H_{2}( 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) = H_{2}(C,S . C) ) & ( for C being Ordinal st C in succ O & C <> 0 & C is limit_ordinal holds

S . C = H_{1}(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) = H_{2}(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> 0 & C is limit_ordinal holds

L0 . C = H_{1}(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) = H_{2}(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> 0 & C is limit_ordinal holds

L0 . C = H_{1}(C,L0 | C) ) ) holds

x1 = x2 ) ) from ORDINAL2:sch 7();

hence ( ex b_{1} being set ex L0 being Sequence st

( b_{1} = 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 b_{1}, b_{2} being set st ex L0 being Sequence st

( b_{1} = 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

( b_{2} = 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

b_{1} = b_{2} ) )
; :: thesis: verum

deffunc H

( 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) = H

S . C = H

( x1 = last L0 & dom L0 = succ O & L0 . 0 = x & ( for C being Ordinal st succ C in succ O holds

L0 . (succ C) = H

L0 . C = H

( x2 = last L0 & dom L0 = succ O & L0 . 0 = x & ( for C being Ordinal st succ C in succ O holds

L0 . (succ C) = H

L0 . C = H

x1 = x2 ) ) from ORDINAL2:sch 7();

hence ( ex b

( b

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 b

( b

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

( b

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

b