
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,
      a being Real
    holds
  (f=F & g=G implies ( G = a*F iff for x be Element of X holds g.x = a*f.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,
      a be Real;
  assume
A1: f=F & g=G;
   reconsider f1=F, g1=G as VECTOR of RAlgebra X by TARSKI:def 3;
   hereby assume
A2:    G = a*F;
     let x be Element of X;
     g1=a*f1 by A2,C0SP1:8;
     hence g.x=a*f.x by A1,FUNCSDOM:4;
   end;
   assume for x be Element of X holds g.x=a*f.x;
   then g1=a*f1 by A1,FUNCSDOM:4;
   hence G =a*F by C0SP1:8;
end;
