reserve x,y,z, X,Y,Z for set,
  n for Element of NAT;
reserve A for set,
  D for non empty set,
  a,b,c,l,r for Element of D,
  o,o9 for BinOp of D,
  f,g,h for Function of A,D;
reserve G for non empty multMagma;

theorem Th21:
  for f1,f2 being Element of .:(G,D), a being Element of D holds (
  f1*f2).a = (f1.a)*(f2.a)
proof
  let f1,f2 be Element of .:(G,D), a be Element of D;
  reconsider g1 = f1, g2 = f2 as Element of Funcs(D, carr(G)) by Th17;
  op(.:(G,D)) = (op(G),carr(G)).:D by Th17;
  then dom (op(G).:(g1,g2)) = D & f1*f2 = op(G).:(g1,g2) by Def2,FUNCT_2:def 1;
  hence thesis by FUNCOP_1:22;
end;
