theorem Th27:
  for X, Y be RealNormSpace for g be Lipschitzian LinearOperator of X,Y
  holds PreNorms(g) is bounded_above
proof
  let X, Y be RealNormSpace;
  let g be Lipschitzian LinearOperator of X,Y;
    consider K be Real such that
A1: 0 <= K and
A2: for x be VECTOR of X holds ||. g.x .|| <= K*||. x .|| by Def8;
    take K;
      let r be ExtReal;
      assume r in PreNorms(g);
      then consider t be VECTOR of X such that
A3:   r=||.g.t.|| and
A4:   ||.t.|| <= 1;
A5:   ||.g.t.|| <= K*||. t .|| by A2;
      K*||. t .|| <= K *1 by A1,A4,XREAL_1:64;
      hence r <=K by A3,A5,XXREAL_0:2;
end;
