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

theorem Th8:
  for x be Element of FT, A be Subset of FT holds x in A^delta iff
  ex y1,y2 being Element of FT st P_1(x,y1,A)=TRUE & P_2(x,y2,A)=TRUE
proof
  let x be Element of FT;
  let A be Subset of FT;
A1: x in A^delta implies ex y1,y2 being Element of FT st P_1(x,y1,A)=TRUE &
  P_2(x,y2,A)=TRUE
  proof
    reconsider z=x as Element of FT;
    assume
A2: x in A^delta;
    then U_FT z meets A by FIN_TOPO:5;
    then consider w1 be object such that
A3: w1 in U_FT z and
A4: w1 in A by XBOOLE_0:3;
    reconsider w1 as Element of FT by A3;
    take w1;
    U_FT z meets A` by A2,FIN_TOPO:5;
    then consider w2 be object such that
A5: w2 in U_FT z & w2 in A` by XBOOLE_0:3;
    take w2;
    thus thesis by A3,A4,A5,Def1,Def2;
  end;
  (ex y1,y2 being Element of FT st P_1(x,y1,A)=TRUE & P_2(x,y2,A)=TRUE)
  implies x in A^delta
  proof
    given y1,y2 being Element of FT such that
A6: P_1(x,y1,A)=TRUE and
A7: P_2(x,y2,A)=TRUE;
    y1 in U_FT x & y1 in A by A6,Def1;
    then U_FT x /\ A <> {} by XBOOLE_0:def 4;
    then
A8: U_FT x meets A;
    y2 in U_FT x & y2 in A` by A7,Def2;
    then U_FT x meets A` by XBOOLE_0:3;
    hence thesis by A8;
  end;
  hence thesis by A1;
end;
