let A be non empty set ; :: thesis: for O1, O2 being Ordinal st O1 c= O2 holds
ConsecutiveSet (A,O1) c= ConsecutiveSet (A,O2)
let O1, O2 be Ordinal; :: thesis: ( O1 c= O2 implies ConsecutiveSet (A,O1) c= ConsecutiveSet (A,O2) )
defpred S1[ Ordinal] means ( O1 c= $1 implies ConsecutiveSet (A,O1) c= ConsecutiveSet (A,$1) );
A1: for O2 being Ordinal st S1[O2] holds
S1[ succ O2]
proof
let O2 be Ordinal; :: thesis: ( S1[O2] implies S1[ succ O2] )
assume A2: ( O1 c= O2 implies ConsecutiveSet (A,O1) c= ConsecutiveSet (A,O2) ) ; :: thesis: S1[ succ O2]
assume A3: O1 c= succ O2 ; :: thesis: ConsecutiveSet (A,O1) c= ConsecutiveSet (A,(succ O2))
per cases ( O1 = succ O2 or O1 <> succ O2 ) ;
end;
end;
A5: for O2 being Ordinal st O2 <> 0 & O2 is limit_ordinal & ( for O3 being Ordinal st O3 in O2 holds
S1[O3] ) holds
S1[O2]
proof
deffunc H1( Ordinal) -> set = ConsecutiveSet (A,$1);
let O2 be Ordinal; :: thesis: ( O2 <> 0 & O2 is limit_ordinal & ( for O3 being Ordinal st O3 in O2 holds
S1[O3] ) implies S1[O2] )
assume that
A6: ( O2 <> 0 & O2 is limit_ordinal ) and
for O3 being Ordinal st O3 in O2 & O1 c= O3 holds
ConsecutiveSet (A,O1) c= ConsecutiveSet (A,O3) ; :: thesis: S1[O2]
consider L being Sequence such that
A7: ( dom L = O2 & ( for O3 being Ordinal st O3 in O2 holds
L . O3 = H1(O3) ) ) from ORDINAL2:sch 2();
 A8: ConsecutiveSet (A,O2) = union (rng L) by A6, A7, Th23;
assume A9: O1 c= O2 ; :: thesis: ConsecutiveSet (A,O1) c= ConsecutiveSet (A,O2)
per cases ( O1 = O2 or O1 <> O2 ) ;
end;
end;
A12: S1[ 0 ] ;
for O2 being Ordinal holds S1[O2] from ORDINAL2:sch 1(A12, A1, A5);
 hence ( O1 c= O2 implies ConsecutiveSet (A,O1) c= ConsecutiveSet (A,O2) ) ; :: thesis: verum