
theorem Th60:
  for L be lower-bounded continuous sup-Semilattice for B be
  with_bottom CLbasis of L holds baseMap B is monotone
proof
  let L be lower-bounded continuous sup-Semilattice;
  let B be with_bottom CLbasis of L;
  set f = baseMap B;
  now
    let x, y be Element of L;
    assume
A1: x <= y;
A2: f.y = waybelow y /\ B by Def12;
    f.x = waybelow x /\ B by Def12;
    then f.x c= f.y by A1,A2,WAYBEL_3:12,XBOOLE_1:26;
    hence f.x <= f.y by YELLOW_1:3;
  end;
  hence thesis by WAYBEL_1:def 2;
end;
