let T be non empty TopSpace; :: thesis: for f being V34 RealMap of T
for p being Point of T holds f . p <= sup f
let f be V34 RealMap of T; :: thesis: for p being Point of T holds f . p <= sup f
set fc = f .: the carrier of T;
A1: f .: the carrier of T is bounded_above by Def12;
set r = sup f;
let p be Point of T; :: thesis: f . p <= sup f
f . p in f .: the carrier of T by FUNCT_2:43;
hence f . p <= sup f by A1, SEQ_4:def 4; :: thesis: verum