let X be non empty set ; :: thesis: for Y being ComplexNormSpace
for g being Function of X, the carrier of Y holds
( g is bounded iff PreNorms g is bounded_above )

let Y be ComplexNormSpace; :: thesis: for g being Function of X, the carrier of Y holds
( g is bounded iff PreNorms g is bounded_above )

let g be Function of X, the carrier of Y; :: thesis: ( g is bounded iff PreNorms g is bounded_above )
now :: thesis: ( PreNorms g is bounded_above implies ex K being Real st g is bounded )
reconsider K = upper_bound () as Real ;
assume A1: PreNorms g is bounded_above ; :: thesis: ex K being Real st g is bounded
A2: 0 <= K
proof
consider r0 being object such that
A3: r0 in PreNorms g by XBOOLE_0:def 1;
reconsider r0 = r0 as Real by A3;
now :: thesis: for r being Real st r in PreNorms g holds
0 <= r
let r be Real; :: thesis: ( r in PreNorms g implies 0 <= r )
assume r in PreNorms g ; :: thesis: 0 <= r
then ex t being Element of X st r = ||.(g . t).|| ;
hence 0 <= r by CLVECT_1:105; :: thesis: verum
end;
then 0 <= r0 by A3;
hence 0 <= K by ; :: thesis: verum
end;
take K = K; :: thesis: g is bounded
now :: thesis: for t being Element of X holds ||.(g . t).|| <= K
let t be Element of X; :: thesis: ||.(g . t).|| <= K
||.(g . t).|| in PreNorms g ;
hence ||.(g . t).|| <= K by ; :: thesis: verum
end;
hence g is bounded by A2; :: thesis: verum
end;
hence ( g is bounded iff PreNorms g is bounded_above ) by Th12; :: thesis: verum