:: deftheorem Def5 defines (+) ORDINAL7:def 5 :
for a, b, b₃ being Ordinal holds
( b₃ = a (+) b iff ex C being Cantor-normal-form Ordinal-Sequence st
( b₃ = Sum^ C & rng (omega -exponent C) = (rng (omega -exponent (CantorNF a))) \/ (rng (omega -exponent (CantorNF b))) & ( for d being object st d in dom C holds
( ( omega -exponent (C . d) in (rng (omega -exponent (CantorNF a))) \ (rng (omega -exponent (CantorNF b))) implies omega -leading_coeff (C . d) = (omega -leading_coeff (CantorNF a)) . (((omega -exponent (CantorNF a)) ") . (omega -exponent (C . d))) ) & ( omega -exponent (C . d) in (rng (omega -exponent (CantorNF b))) \ (rng (omega -exponent (CantorNF a))) implies omega -leading_coeff (C . d) = (omega -leading_coeff (CantorNF b)) . (((omega -exponent (CantorNF b)) ") . (omega -exponent (C . d))) ) & ( omega -exponent (C . d) in (rng (omega -exponent (CantorNF a))) /\ (rng (omega -exponent (CantorNF b))) implies omega -leading_coeff (C . d) = ((omega -leading_coeff (CantorNF a)) . (((omega -exponent (CantorNF a)) ") . (omega -exponent (C . d)))) + ((omega -leading_coeff (CantorNF b)) . (((omega -exponent (CantorNF b)) ") . (omega -exponent (C . d)))) ) ) ) ) );