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

end;

then 0 <= r0 by A3;
hence 0 <= K by A1, A3, SEQ_4:def 1; :: 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 A1, SEQ_4:def 1; :: thesis: verum

end;

hence g is bounded by A2; :: thesis: verum

end;

hence ( g is bounded iff PreNorms g is bounded_above ) by Th11; :: thesis: verum