
theorem Th17:
  for X be non empty set, Y be ComplexNormSpace, f being Point of
C_NormSpace_of_BoundedFunctions(X,Y), g be bounded Function of X,the carrier of
  Y st g=f holds for t be Element of X holds ||.g.t.|| <= ||.f.||
proof
  let X be non empty set;
  let Y be ComplexNormSpace;
  let f being Point of C_NormSpace_of_BoundedFunctions(X,Y);
  let g be bounded Function of X,the carrier of Y such that
A1: g=f;
A2: PreNorms(g) is non empty bounded_above by Th12;
  now
    let t be Element of X;
    ||.g.t.|| in PreNorms(g);
    then ||.g.t.|| <= upper_bound PreNorms(g) by A2,SEQ_4:def 1;
    then ||.g.t.|| <= ComplexBoundedFunctionsNorm(X,Y).g by Th15;
    hence ||.g.t.|| <= ||.f.|| by A1;
  end;
  hence thesis;
end;
