reserve FT for non empty RelStr;
reserve A for Subset of FT;

theorem
  for x be Element of FT, A be Subset of FT holds x in A^b iff ex y1
  being Element of FT st P_1(x,y1,A)=TRUE
proof
  let x be Element of FT;
  let A be Subset of FT;
A1: x in A^b implies ex y1 being Element of FT st P_1(x,y1,A)=TRUE
  proof
    reconsider z=x as Element of FT;
    assume x in A^b;
    then U_FT z meets A by FIN_TOPO:8;
    then consider w be object such that
A2: w in U_FT z and
A3: w in A by XBOOLE_0:3;
    reconsider w as Element of FT by A2;
    take w;
    thus thesis by A2,A3,Def1;
  end;
  (ex y1 being Element of FT st P_1(x,y1,A)=TRUE) implies x in A^b
  proof
    given y be Element of FT such that
A4: P_1(x,y,A)=TRUE;
    y in U_FT x & y in A by A4,Def1;
    then y in (U_FT x /\ A) by XBOOLE_0:def 4;
    then U_FT x meets A;
    hence thesis;
  end;
  hence thesis by A1;
end;
