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^n iff P_A(x,A)=
  TRUE & 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^n implies P_A(x,A)=TRUE & ex y being Element of FT st P_1(x,y,A)=
  TRUE & P_e(x,y)=FALSE
  proof
    assume
A2: x in A^n;
    then (U_FT x \ {x}) meets A by FIN_TOPO:10;
    then (U_FT x \ {x}) /\ A <> {};
    then consider y being Element of FT such that
A3: y in ((U_FT x \ {x}) /\ A) by SUBSET_1:4;
A4: y in U_FT x \ {x} by A3,XBOOLE_0:def 4;
    then
A5: y in U_FT x by XBOOLE_0:def 5;
    not y in {x} by A4,XBOOLE_0:def 5;
    then not x = y by TARSKI:def 1;
    then
A6: P_e(x,y)=FALSE by Def5;
    y in A by A3,XBOOLE_0:def 4;
    then
A7: P_1(x,y,A)=TRUE by A5,Def1;
    x in A by A2,FIN_TOPO:10;
    hence thesis by A6,A7,Def4;
  end;
  (P_A(x,A)=TRUE & ex y being Element of FT st P_1(x,y,A)=TRUE & P_e(x,y)
  =FALSE ) implies x in A^n
  proof
    assume that
A8: P_A(x,A)=TRUE and
A9: ex y being Element of FT st P_1(x,y,A)=TRUE & P_e(x,y)=FALSE;
    consider y being Element of FT such that
A10: P_1(x,y,A)=TRUE and
A11: P_e(x,y)=FALSE by A9;
    x <> y by A11,Def5;
    then
A12: not y in {x} by TARSKI:def 1;
    y in U_FT x by A10,Def1;
    then
A13: y in (U_FT x \ {x}) by A12,XBOOLE_0:def 5;
    y in A by A10,Def1;
    then
A14: (U_FT x \ {x}) meets A by A13,XBOOLE_0:3;
    x in A by A8,Def4;
    hence thesis by A14,FIN_TOPO:10;
  end;
  hence thesis by A1;
end;
