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 Th30:
  f=F & g=G implies ( G = a*F iff for x be Element of X holds g.x = a*f.x )
proof
  reconsider f1=F, g1=G as VECTOR of R_Algebra_of_BoundedFunctions X;
A1: G=a*F iff g1=a*f1;
  assume f=F & g=G;
  hence thesis by A1,Th13;
end;
