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^s iff P_A(x,A)=
  TRUE & not(ex y being Element of FT st P_1(x,y,A)=TRUE & P_e(x,y)=FALSE )
proof
  let x be Element of FT;
  let A be Subset of FT;
A1: x in A^s implies P_A(x,A)=TRUE & not(ex y being Element of FT st P_1(x,y
  ,A)=TRUE & P_e(x,y)=FALSE )
  proof
    assume
A2: x in A^s;
    then (U_FT x \ {x}) misses A by FIN_TOPO:9;
    then
A3: (U_FT x \ {x}) /\ A = {};
A4: not(ex y being Element of FT st P_1(x,y,A)=TRUE & P_e(x,y)=FALSE )
    proof
      given y being Element of FT such that
A5:   P_1(x,y,A)=TRUE and
A6:   P_e(x,y)=FALSE;
      not x = y by A6,Def5;
      then
A7:   not y in {x} by TARSKI:def 1;
      y in U_FT x by A5,Def1;
      then
A8:   y in (U_FT x \ {x}) by A7,XBOOLE_0:def 5;
      y in A by A5,Def1;
      hence contradiction by A3,A8,XBOOLE_0:def 4;
    end;
    x in A by A2,FIN_TOPO:9;
    hence thesis by A4,Def4;
  end;
  P_A(x,A)=TRUE & not(ex y being Element of FT st P_1(x,y,A)=TRUE & P_e(x
  ,y)=FALSE ) implies x in A^s
  proof
    assume that
A9: P_A(x,A)=TRUE and
A10: not(ex y being Element of FT st P_1(x,y,A)=TRUE & P_e(x,y)=FALSE );
    for y be Element of FT holds not y in ((U_FT x \ {x}) /\ A)
    proof
      let y be Element of FT;
      not (P_1(x,y,A)=TRUE & P_e(x,y)=FALSE) by A10;
      then not( y in U_FT x & (not x = y) & y in A ) by Def1,Def5;
      then not( y in U_FT x & (not y in {x}) & y in A ) by TARSKI:def 1;
      then not(y in (U_FT x \ {x}) & y in A ) by XBOOLE_0:def 5;
      hence thesis by XBOOLE_0:def 4;
    end;
    then (U_FT x \ {x}) /\ A = {} by SUBSET_1:4;
    then
A11: (U_FT x \ {x}) misses A;
    x in A by A9,Def4;
    hence thesis by A11;
  end;
  hence thesis by A1;
end;
