reserve e for set;

theorem
  for I1 being set, I2 being non empty set, f being Function of I1,I2, B
  ,C being ManySortedSet of I2, G being ManySortedFunction of B,C holds G*f is
  ManySortedFunction of B*f, C*f
proof
  let I1 be set, I2 be non empty set, f be Function of I1,I2, B,C be
  ManySortedSet of I2, G be ManySortedFunction of B,C;
  let i be object;
  assume
A1: i in I1;
  then
A2: G.(f.i) is Function of B.(f.i),C.(f.i) by FUNCT_2:5,PBOOLE:def 15;
A3: i in dom f by A1,FUNCT_2:def 1;
  then B.(f.i) = (B*f).i & C.(f.i) = (C*f).i by FUNCT_1:13;
  hence thesis by A3,A2,FUNCT_1:13;
end;
