reserve f, g, h for Function;
reserve x, y, z, u, X for set,
  A for non empty set,
  n for Element of NAT,
  f for Function of X, X;
reserve f for c=-monotone Function of bool X, bool X,
  S for Subset of X;
reserve X, Y for non empty set,
  f for Function of X, Y,
  g for Function of Y, X;
reserve L for Lattice,
  f for Function of the carrier of L, the carrier of L,
  x for Element of L,
  O, O1, O2, O3, O4 for Ordinal,
  T for Sequence;
reserve L for complete Lattice,
  f for monotone UnOp of L,
  a, b for Element of L;
reserve f for monotone UnOp of BooleLatt A;

theorem Th46:
  for f being UnOp of BooleLatt A holds f is monotone iff f is c=-monotone
proof
  let f be UnOp of BooleLatt A;
  thus f is monotone implies f is c=-monotone
  proof
    assume
A1: f is monotone;
    let x, y be Element of BooleLatt A;
    assume x c= y;
    then x [= y by LATTICE3:2;
    then f.x [= f.y by A1;
    hence thesis by LATTICE3:2;
  end;
  assume
A2: f is c=-monotone;
  let p, q be Element of BooleLatt A;
  assume p [= q;
  then p c= q by LATTICE3:2;
  then f.p c= f.q by A2;
  hence f.p [= f.q by LATTICE3:2;
end;
