reserve x, X, Y for set;

theorem
  for L, M being non empty RelStr for f being Function of L, M for X
  being Subset of L, x being Element of L holds f is monotone & x is_<=_than X
  implies f.x is_<=_than f.:X
proof
  let L1,L2 be non empty RelStr, g be Function of L1,L2;
  let X be Subset of L1, x be Element of L1 such that
A1: g is monotone and
A2: x is_<=_than X;
  let y2 be Element of L2;
  assume y2 in g.:X;
  then consider x2 being Element of L1 such that
A3: x2 in X and
A4: y2 = g.x2 by FUNCT_2:65;
  reconsider x2 as Element of L1;
  x <= x2 by A2,A3;
  hence thesis by A1,A4;
end;
