reserve i, x, I for set,
  A, B, M for ManySortedSet of I,
  f, f1 for Function;
reserve SF, SG for SubsetFamily of M;
reserve E, T for Element of Bool M;
reserve g, h for SetOp of M;

theorem :: CLOSURE:18
  for A being Element of Bool M st A = E (/\) T holds g is monotonic
  implies g.A c= g.E (/\) g.T
proof
  let A be Element of Bool M such that
A1: A = E (/\) T;
  assume
A2: g is monotonic;
  E (/\) T c= T by PBOOLE:15;
  then
A3: g.A c= g.T by A1,A2;
  E (/\) T c= E by PBOOLE:15;
  then g.A c= g.E by A1,A2;
  hence thesis by A3,PBOOLE:17;
end;
