
theorem
  for m be non zero Element of NAT, k be Element of NAT,
      X be non empty open Subset of REAL m,
      F,G,H being VECTOR of R_Algebra_of_Ck_Functions(k,X),
      f,g,h being PartFunc of REAL m, REAL
    holds
     (f=F & g=G & h=H implies ( H = F*G iff (for x be Element of X
        holds h.x = f.x * g.x)))
proof
  let m be non zero Element of NAT, k be Element of NAT,
      X be non empty open Subset of REAL m,
      F,G,H be VECTOR of R_Algebra_of_Ck_Functions(k,X),
      f,g,h be PartFunc of REAL m, REAL;
  assume
A1: f=F & g=G & h=H;
   reconsider f1=F, g1=G, h1=H as VECTOR of RAlgebra X by TARSKI:def 3;
   hereby assume
A2:  H = F*G;
     let x be Element of X;
     h1 = f1*g1 by A2,C0SP1:8;
     hence h.x = f.x * g.x by A1,FUNCSDOM:2;
   end;
   assume for x be Element of X holds h.x = f.x * g.x;
   then h1 = f1 * g1 by A1,FUNCSDOM:2;
   hence H = F * G by C0SP1:8;
end;
