reserve X for non empty set;
reserve x for Element of X;
reserve d1,d2 for Element of X;
reserve A for BinOp of X;
reserve M for Function of [:X,X:],X;
reserve V for Ring;
reserve V1 for Subset of V;
reserve V for Algebra;
reserve V1 for Subset of V;
reserve MR for Function of [:REAL,X:],X;
reserve a for Real;
reserve F,G,H for VECTOR of R_Algebra_of_BoundedFunctions X;
reserve f,g,h for Function of X,REAL;
reserve F,G,H for Point of R_Normed_Algebra_of_BoundedFunctions X;

theorem Th29:
  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,Th12;
end;
