reserve x,y,z for Real,
  R for real non empty RelStr,
  a,b for Element of R;

theorem Th15:
  for R being complete Heyting LATTICE, X being Subset of R, y be
Element of R holds "\/"(X,R) "/\" y = "\/"({x "/\" y where x is Element of R: x
  in X},R)
proof
  let R be complete Heyting LATTICE, X be Subset of R, y be Element of R;
  set Z = {y "/\" x where x is Element of R: x in X}, W = {x "/\" y where x is
  Element of R: x in X};
A1: W c= Z
  proof
    let w be object;
    assume w in W;
    then ex x being Element of R st w = x "/\" y & x in X;
    hence thesis;
  end;
  Z c= W
  proof
    let z be object;
    assume z in Z;
    then ex x being Element of R st z = y "/\" x & x in X;
    hence thesis;
  end;
  then
  ( for z being Element of R holds z "/\" is lower_adjoint & ex_sup_of X,R
  )& Z = W by A1,WAYBEL_1:def 19,YELLOW_0:17;
  hence thesis by WAYBEL_1:63;
end;
