defpred S₁[ Ordinal] means not ConsecutiveSet A,A is empty ;
A1: S₁[ {} ] by Th24;
A2: for O being Ordinal st S₁[O] holds
S₁[ succ O]

proof

let O1 be Ordinal; :: thesis: ( S₁[O1] implies S₁[ succ O1] )
assume not ConsecutiveSet A,O1 is empty ; :: thesis: S₁[ succ O1]
ConsecutiveSet A,(succ O1) = new_set (ConsecutiveSet A,O1) by Th25;
hence S₁[ succ O1] ; :: thesis: verum

end;

A3: for O being Ordinal st O <> {} & O is limit_ordinal & ( for B being Ordinal st B in O holds
S₁[B] ) holds
S₁[O]

proof

let O1 be Ordinal; :: thesis: ( O1 <> {} & O1 is limit_ordinal & ( for B being Ordinal st B in O1 holds
S₁[B] ) implies S₁[O1] )
assume A4: ( O1 <> {} & O1 is limit_ordinal & ( for O2 being Ordinal st O2 in O1 holds
not ConsecutiveSet A,O2 is empty ) ) ; :: thesis: S₁[O1]
deffunc H₁( Ordinal) -> set = ConsecutiveSet A,A;
consider Ls being T-Sequence such that
A5: ( dom Ls = O1 & ( for O2 being Ordinal st O2 in O1 holds
Ls . O2 = H₁(O2) ) ) from ORDINAL2:sch 2();
A6: ConsecutiveSet A,O1 = union (rng Ls) by A4, A5, Th26;
A7: {} in O1 by A4, ORDINAL3:10;
then Ls . {} = ConsecutiveSet A,{} by A5
.= A by Th24 ;
then A in rng Ls by A5, A7, FUNCT_1:def 5;
then A8: A c= ConsecutiveSet A,O1 by A6, ZFMISC_1:92;
thus S₁[O1] by A8; :: thesis: verum

end;

for O being Ordinal holds S₁[O] from ORDINAL2:sch 1(A1, A2, A3);
hence not ConsecutiveSet A,O is empty ; :: thesis: verum