
theorem Th18:
  for X be non empty set for Y be RealNormSpace for f being Point
  of R_NormSpace_of_BoundedFunctions(X,Y) st f = 0.
  R_NormSpace_of_BoundedFunctions(X,Y) holds 0 = ||.f.||
proof
  let X be non empty set;
  let Y be RealNormSpace;
  let f being Point of R_NormSpace_of_BoundedFunctions(X,Y) such that
A1: f = 0.R_NormSpace_of_BoundedFunctions(X,Y);
  thus ||.f.|| = 0
  proof
    reconsider g=f as bounded Function of X, the carrier of Y by Def5;
    set z = X --> 0.Y;
    reconsider z as Function of X, the carrier of Y;
    consider r0 be object such that
A2: r0 in PreNorms(g) by XBOOLE_0:def 1;
    reconsider r0 as Real by A2;
A3: (for s be Real st s in PreNorms(g) holds s <= 0) implies upper_bound
    PreNorms(g) <= 0 by SEQ_4:45;
A4: PreNorms(g) is non empty bounded_above by Th11;
A5: z=g by A1,Th15;
A6: now
      let r be Real;
      assume r in PreNorms(g);
      then consider t be Element of X such that
A7:   r=||.g.t.||;
      ||.g.t.|| = ||.0.Y.|| by A5,FUNCOP_1:7
        .= 0;
      hence 0 <= r & r <=0 by A7;
    end;
    then 0 <= r0 by A2;
    then upper_bound PreNorms(g) = 0 by A6,A4,A2,A3,SEQ_4:def 1;
    hence thesis by Th14;
  end;
end;
