deffunc H₁( Ordinal) -> set = f,$1 +. a;
defpred S₁[ Ordinal] means f,$1 +. a is Element of L;

A1: now

let O1 be Ordinal; :: thesis: ( S₁[O1] implies S₁[ succ O1] )
assume S₁[O1] ; :: thesis: S₁[ succ O1]
then reconsider fa = f,O1 +. a as Element of L ;
f . fa = f,(succ O1) +. a by Th18;
hence S₁[ succ O1] ; :: thesis: verum

end;

A2: now

let O1 be Ordinal; :: thesis: ( O1 <> {} & O1 is limit_ordinal & ( for O2 being Ordinal st O2 in O1 holds
S₁[O2] ) implies S₁[O1] )
assume that
A3: ( O1 <> {} & O1 is limit_ordinal ) and
for O2 being Ordinal st O2 in O1 holds
S₁[O2] ; :: thesis: S₁[O1]
consider Ls being T-Sequence such that
A4: ( dom Ls = O1 & ( for O2 being Ordinal st O2 in O1 holds
Ls . O2 = H₁(O2) ) ) from ORDINAL2:sch 2();
f,O1 +. a = "\/" (rng Ls),L by A3, A4, Th20;
hence S₁[O1] ; :: thesis: verum

end;

A5: S₁[ {} ] by Th16;
for O being Ordinal holds S₁[O] from ORDINAL2:sch 1(A5, A1, A2);
hence f,O +. a is Element of L ; :: thesis: verum