
theorem Th8:
  for FT being filled non empty RelStr, A,B being Subset of FT st
  FT is symmetric & A^b misses B holds A misses (B^b)
proof
  let FT be filled non empty RelStr, A,B be Subset of FT;
  assume that
A1: FT is symmetric and
A2: A^b misses B;
  now
    assume A meets (B^b);
    then consider x being object such that
A3: x in A and
A4: x in B^b by XBOOLE_0:3;
    consider x2 being Element of FT such that
A5: x2=x and
A6: U_FT x2 meets B by A4;
    set y = the Element of U_FT x2 /\ B;
A7: U_FT x2 /\ B <>{} by A6,XBOOLE_0:def 7;
    then
A8: y in U_FT x2 by XBOOLE_0:def 4;
    then reconsider y2=y as Element of FT;
    x2 in U_FT y2 by A1,A8;
    then x2 in U_FT y2 /\ A by A3,A5,XBOOLE_0:def 4;
    then U_FT y2 meets A by XBOOLE_0:def 7;
    then
A9: y2 in A^b;
    y in B by A7,XBOOLE_0:def 4;
    hence contradiction by A2,A9,XBOOLE_0:3;
  end;
  hence thesis;
end;
