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^i iff for y be
  Element of FT holds (P_0(x,y)=TRUE implies P_1(x,y,A)=TRUE)
proof
  let x be Element of FT;
  let A be Subset of FT;
A1: (for y be Element of FT holds (P_0(x,y)=TRUE implies P_1(x,y,A)=TRUE))
  implies x in A^i
  proof
    assume
A2: for y be Element of FT holds (P_0(x,y)=TRUE implies P_1(x,y,A)= TRUE);
    for y be Element of FT holds y in (U_FT x) implies y in U_FT x & y in A
    proof
      let y be Element of FT;
      assume y in (U_FT x);
      then P_0(x,y)=TRUE by Def3;
      then P_1(x,y,A)=TRUE by A2;
      hence thesis by Def1;
    end;
    then for y be Element of FT holds y in (U_FT x) implies y in A;
    then (U_FT x) c= A;
    hence thesis;
  end;
  x in A^i implies for y be Element of FT holds (P_0(x,y)=TRUE implies P_1
  (x,y,A)=TRUE)
  proof
    assume x in A^i;
    then
A3: U_FT x c= A by FIN_TOPO:7;
    let y be Element of FT;
    assume P_0(x,y)=TRUE;
    then y in U_FT x by Def3;
    hence thesis by A3,Def1;
  end;
  hence thesis by A1;
end;
