reserve T for non empty RelStr,
  a for Element of T;
reserve a for set;
reserve P for non empty POSet_set,
  A,B for Element of P;

theorem Th6:
  for A,B,C be non empty RelStr for f,g be Function st f in
  MonFuncs (A,B) & g in MonFuncs (B,C) holds (g*f) in MonFuncs (A,C)
proof
  let A,B,C be non empty RelStr;
  let f,g be Function;
  assume that
A1: f in MonFuncs (A,B) and
A2: g in MonFuncs (B,C);
  consider f9 be Function of A, B such that
A3: f = f9 and
  f9 in Funcs (the carrier of A, the carrier of B) and
A4: f9 is monotone by A1,Def6;
  consider g9 be Function of B, C such that
A5: g = g9 and
  g9 in Funcs (the carrier of B, the carrier of C) and
A6: g9 is monotone by A2,Def6;
  consider gf be Function of A,C such that
A7: gf = g9 * f9 & gf is monotone by A4,A6,Lm1;
  gf in Funcs (the carrier of A, the carrier of C) by FUNCT_2:8;
  hence thesis by A3,A5,A7,Def6;
end;
