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^deltai 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) =
  TRUE
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) = TRUE implies x in A^deltai
  proof
    assume ( ex y1,y2 being Element of FT st P_1(x,y1,A)=TRUE & P_2(x,y2,A)=
    TRUE)& P_A(x, A) = TRUE;
    then x in A^delta & x in A by Def4,Th8;
    hence thesis by XBOOLE_0:def 4;
  end;
  x in A^deltai 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) = TRUE
  proof
    assume x in A^deltai;
    then x in A & x in A^delta by XBOOLE_0:def 4;
    hence thesis by Def4,Th8;
  end;
  hence thesis by A1;
end;
