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^f iff ex y
  being Element of FT st P_A(y,A)=TRUE & P_0(y,x)=TRUE
proof
  let x be Element of FT;
  let A be Subset of FT;
A1: (ex y being Element of FT st P_A(y,A)=TRUE & P_0(y,x)=TRUE) implies x in
  A^f
  proof
    assume ex y being Element of FT st P_A(y,A)=TRUE & P_0(y,x)=TRUE;
    then consider y being Element of FT such that
A2: P_A(y,A)=TRUE & P_0(y,x)=TRUE;
    y in A & x in U_FT y by A2,Def3,Def4;
    hence thesis;
  end;
  x in A^f implies ex y being Element of FT st P_A(y,A)=TRUE & P_0(y,x)= TRUE
  proof
    assume x in A^f;
    then consider y being Element of FT such that
A3: y in A & x in U_FT y by FIN_TOPO:11;
    P_A(y,A)=TRUE & P_0(y,x)=TRUE by A3,Def3,Def4;
    hence thesis;
  end;
  hence thesis by A1;
end;
