
theorem Th11:
  for X be non empty set for Y be RealNormSpace for g be bounded
  Function of X,the carrier of Y holds PreNorms(g) is bounded_above
proof
  let X be non empty set;
  let Y be RealNormSpace;
  let g be bounded Function of X,the carrier of Y;
    consider K be Real such that
    0 <= K and
A1: for x be Element of X holds ||. g.x .|| <= K by Def4;
    take K;
      let r be ExtReal;
      assume r in PreNorms(g);
      then ex t be Element of X st r=||.g.t.||;
      hence r <=K by A1;
end;
