defpred S₁[ Ordinal] means not ConsecutiveSet2 A,A is empty ;
A1: for O1 being Ordinal st S₁[O1] holds
S₁[ succ O1]

proof

let O1 be Ordinal; :: thesis: ( S₁[O1] implies S₁[ succ O1] )
assume not ConsecutiveSet2 A,O1 is empty ; :: thesis: S₁[ succ O1]
ConsecutiveSet2 A,(succ O1) = new_set2 (ConsecutiveSet2 A,O1) by Th16;
hence S₁[ succ O1] ; :: thesis: verum

end;

A2: for O1 being Ordinal st O1 <> {} & O1 is limit_ordinal & ( for O2 being Ordinal st O2 in O1 holds
S₁[O2] ) holds
S₁[O1]

proof

deffunc H₁( Ordinal) -> set = ConsecutiveSet2 A,A;
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 <> {} and
A4: O1 is limit_ordinal and
for O2 being Ordinal st O2 in O1 holds
not ConsecutiveSet2 A,O2 is empty ; :: thesis: S₁[O1]
A5: {} in O1 by A3, ORDINAL3:10;
consider Ls being T-Sequence such that
A6: ( dom Ls = O1 & ( for O2 being Ordinal st O2 in O1 holds
Ls . O2 = H₁(O2) ) ) from ORDINAL2:sch 2();
Ls . {} = ConsecutiveSet2 A,{} by A3, A6, ORDINAL3:10
.= A by Th15 ;
then A7: A in rng Ls by A6, A5, FUNCT_1:def 5;
ConsecutiveSet2 A,O1 = union (rng Ls) by A3, A4, A6, Th17;
then A c= ConsecutiveSet2 A,O1 by A7, ZFMISC_1:92;
hence S₁[O1] ; :: thesis: verum

end;

A8: S₁[ {} ] by Th15;
for O being Ordinal holds S₁[O] from ORDINAL2:sch 1(A8, A1, A2);
hence not ConsecutiveSet2 A,O is empty ; :: thesis: verum