reserve x, X, Y for set;

theorem
  for L, M, N being non empty RelStr for f being Function of L, M, g
  being Function of M, N holds f is monotone & g is monotone implies g*f is
  monotone
proof
  let L1,L2,L3 be non empty RelStr;
  let g1 be Function of L1,L2, g2 be Function of L2,L3 such that
A1: g1 is monotone and
A2: g2 is monotone;
  let s1,s2 be Element of L1;
  assume s1 <= s2;
  then g1.s1 <= g1.s2 by A1;
  then g2.(g1.s1) <= g2.(g1.s2) by A2;
  then (g2*g1).s1 <= g2.(g1.s2) by FUNCT_2:15;
  hence thesis by FUNCT_2:15;
end;
