theorem Th31:
  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
  reconsider f1=F, g1=G, h1=H as VECTOR of R_Algebra_of_BoundedFunctions X;
A1: H=F*G iff h1=f1*g1;
  assume f=F & g=G & h=H;
  hence thesis by A1,Th14;
end;
