let A be Ordinal; :: thesis: ( A is limit_ordinal iff for B being Ordinal
for n being Nat st B in A holds
B +^ n in A )
thus ( A is limit_ordinal implies for B being Ordinal
for n being Nat st B in A holds
B +^ n in A ) :: thesis: ( ( for B being Ordinal
for n being Nat st B in A holds
B +^ n in A ) implies A is limit_ordinal )

proof

assume A1: A is limit_ordinal ; :: thesis: for B being Ordinal
for n being Nat st B in A holds
B +^ n in A
let B be Ordinal; :: thesis: for n being Nat st B in A holds
B +^ n in A
let n be Nat; :: thesis: ( B in A implies B +^ n in A )
defpred S₁[ Nat] means B +^ $1 in A;
assume B in A ; :: thesis: B +^ n in A
then A2: S₁[ 0 ] by ORDINAL2:27;
A3: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
Segm (k + 1) = succ (Segm k) by NAT_1:38;
then B +^ (k + 1) = succ (B +^ k) by ORDINAL2:28;
hence ( S₁[k] implies S₁[k + 1] ) by A1, ORDINAL1:28; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A2, A3);
hence B +^ n in A ; :: thesis: verum

end;

assume A4: for B being Ordinal
for n being Nat st B in A holds
B +^ n in A ; :: thesis: A is limit_ordinal

now :: thesis: for B being Ordinal st B in A holds
succ B in A

let B be Ordinal; :: thesis: ( B in A implies succ B in A )
assume B in A ; :: thesis: succ B in A
then B +^ 1 in A by A4;
hence succ B in A by ORDINAL2:31; :: thesis: verum

end;

hence A is limit_ordinal by ORDINAL1:28; :: thesis: verum