reserve X for non empty set;
reserve Y for RealLinearSpace;
reserve f,g,h for Element of Funcs(X,the carrier of Y);
reserve a,b for Real;
reserve u,v,w for VECTOR of RLSStruct(#Funcs(X,the carrier of Y), (FuncZero(X,
    Y)),FuncAdd(X,Y),FuncExtMult(X,Y)#);

theorem
  for X be non empty set for Y be RealLinearSpace for f,h be VECTOR of
RealVectSpace(X,Y) for a be Real holds h = a*f iff for x be Element of X holds
  h.x = a * f.x by Th2;
