reserve x,y for object,
        D,D1,D2 for non empty set,
        i,j,k,m,n for Nat,
        f,g for FinSequence of D*,
        f1 for FinSequence of D1*,
        f2 for FinSequence of D2*;
reserve f for complex-valued Function,
        g,h for complex-valued FinSequence;

theorem Th41:
  for f be Function-yielding Function, g be Function holds
    (g*.f).x = g*(f.x)
    proof
      let f be Function-yielding Function, g be Function;
      per cases;
      suppose x in dom f;
        hence thesis by FOMODEL2:def 6;
      end;
      suppose not x in dom f;
        then not x in dom (g*.f) & f.x={} by FOMODEL2:def 6,FUNCT_1:def 2;
        then (g*.f).x={} & g*(f.x)={} by FUNCT_1:def 2;
        hence thesis;
      end;
    end;
