reserve n for non zero Element of NAT;
reserve a,b,r,t for Real;

theorem
 for X be non empty closed_interval Subset of REAL,
     Y be RealNormSpace,
     f,h be Point of R_NormSpace_of_ContinuousFunctions(X,Y),
     f9,h9 be continuous PartFunc of REAL,Y
   st f9=f & h9=h & dom f9=X & dom h9=X
     holds (h = a*f iff for x be Element of X
  holds h9/.x = a*f9/.x)
proof
  let X be non empty closed_interval Subset of REAL,
  Y be RealNormSpace,
  f,h be Point of R_NormSpace_of_ContinuousFunctions(X,Y);
  reconsider f1=f, h1=h as VECTOR of R_VectorSpace_of_ContinuousFunctions(X,Y);
  let f9,h9 be continuous PartFunc of REAL,Y;
  assume
A1: f9=f & h9=h & dom f9=X & dom h9=X;
  h=a*f iff h1=a*f1;
  hence thesis by A1,Th11;
end;
