reserve x, y for object, I for set,
  A, B, X, Y for ManySortedSet of I;

theorem :: LATTICE4:1
  union Y c= A & X in Y implies X c= A
proof
  assume that
A1: union Y c= A and
A2: X in Y;
  let i be object;
  assume
A3: i in I;
  then (union Y).i c= A.i by A1;
  then
A4: union (Y.i) c= A.i by A3,Def2;
  X.i in Y.i by A2,A3;
  hence thesis by A4,SETFAM_1:41;
end;
