
theorem Th31:
  for X, x being set, F being Dependency-set of X holds x in
  saturated-subsets F iff ex B, A being Subset of X st x = B & A ^|^ B, F
proof
  let X, x be set, F be Dependency-set of X;
  hereby
    assume x in saturated-subsets F;
    then consider B being Subset of X such that
A1: x = B and
A2: ex A being Subset of X st A^|^B,F;
    consider A being Subset of X such that
A3: A^|^B,F by A2;
    take B, A;
    thus x = B & A^|^B,F by A1,A3;
  end;
  given B, A being Subset of X such that
A4: x = B and
A5: A ^|^ B, F;
  thus thesis by A4,A5;
end;
