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^deltao iff (ex
y1,y2 being Element of FT st P_1(x,y1,A)=TRUE & P_2(x,y2,A)=TRUE) & P_A(x,A) =
  FALSE
proof
  let x be Element of FT;
  let A be Subset of FT;
A1: (ex y1,y2 being Element of FT st P_1(x,y1,A)=TRUE & P_2(x,y2,A)=TRUE) &
  P_A(x,A) = FALSE implies x in A^deltao
  proof
    assume that
A2: ex y1,y2 being Element of FT st P_1(x,y1,A)=TRUE & P_2(x,y2,A)= TRUE and
A3: P_A(x,A) = FALSE;
    not x in A by A3,Def4;
    then
A4: x in A` by XBOOLE_0:def 5;
    x in A^delta by A2,Th8;
    hence thesis by A4,XBOOLE_0:def 4;
  end;
  x in A^deltao implies (ex y1,y2 being Element of FT st P_1(x,y1,A)=TRUE
  & P_2(x,y2,A)=TRUE) & P_A(x,A) = FALSE
  proof
    assume
A5: x in A^deltao;
    then x in A` by XBOOLE_0:def 4;
    then
A6: not x in A by XBOOLE_0:def 5;
    x in A^delta by A5,XBOOLE_0:def 4;
    hence thesis by A6,Def4,Th8;
  end;
  hence thesis by A1;
end;
