 reserve U for set,
         X, Y for Subset of U;

theorem Th1:
  for x being set holds x in Inter (X,Y) iff X c= x & x c= Y
  proof
    let x be set;
    hereby assume x in Inter (X,Y); then
      consider A9 being Element of bool U such that
A1:   x = A9 & X c= A9 & A9 c= Y;
      thus X c= x & x c= Y by A1;
    end;
    assume
A2: X c= x & x c= Y; then
    x is Subset of U by XBOOLE_1:1;
    hence thesis by A2;
  end;
