reserve GX,GY for non empty TopSpace,
  x,y for Point of GX,
  r,s for Real;
reserve Q,P1,P2 for Subset of TOP-REAL 2;
reserve P for Subset of TOP-REAL 2;
reserve w1,w2 for Point of TOP-REAL 2;
reserve pa,pb for Point of TOP-REAL 2,
  s1,t1,s2,t2,s,t,s3,t3,s4,t4,s5,t5,s6,t6, l,sa,sd,ta,td for Real;
reserve s1a,t1a,s2a,t2a,s3a,t3a,sb,tb,sc,tc for Real;

theorem
  for S being Subset of TOP-REAL 2 st S is Jordan holds S`<> {} &
  ex A1,A2 being Subset of TOP-REAL 2 st
  ex C1,C2 being Subset of (TOP-REAL 2)|S` st S` = A1 \/ A2 & A1 misses A2 &
  (Cl A1) \ A1 = (Cl A2) \ A2 & C1 = A1 & C2 = A2 &
  C1 is a_component & C2 is a_component &
  for C3 being Subset of (TOP-REAL 2)|S` st C3 is a_component holds
  C3 = C1 or C3 = C2
proof
  let S be Subset of TOP-REAL 2;
  assume
A1: S is Jordan;
  then reconsider S9 = S` as non empty Subset of TOP-REAL 2;
  consider A1,A2 being Subset of TOP-REAL 2 such that
A2: S`=A1 \/ A2 and
A3: A1 misses A2 and
A4: (Cl A1)\A1=(Cl A2)\A2 and
A5: for C1,C2 being Subset of (TOP-REAL 2)|S`
  st C1=A1 & C2=A2 holds C1 is a_component & C2 is a_component by A1;
A6: A1 c= S` by A2,XBOOLE_1:7;
A7: A2 c= S` by A2,XBOOLE_1:7;
A8: [#]((TOP-REAL 2)|S`)=S` by PRE_TOPC:def 5;
  A1 c= [#]((TOP-REAL 2)|S`) by A6,PRE_TOPC:def 5;
  then reconsider G0A=A1,G0B=A2 as Subset of (TOP-REAL 2)|S9 by A7,
PRE_TOPC:def 5;
A9: G0A=A1;
  G0B=A2;
  then
A10: G0A is a_component by A5;
A11: G0B is a_component by A5,A9;
  now
    let C3 be Subset of (TOP-REAL 2)|S9;
    assume
A12: C3 is a_component;
    then
A13: C3 <>{}((TOP-REAL 2)|S9) by CONNSP_1:32;
    C3 /\(G0A \/ G0B)=C3 by A2,A8,XBOOLE_1:28;
    then
A14: (C3 /\ G0A) \/ (C3 /\ G0B) <>{} by A13,XBOOLE_1:23;
    now per cases by A14;
      suppose C3 /\ G0A<>{};
        then
A15:    C3 meets G0A;
A16:    C3 is connected by A12;
A17:    G0A is connected by A10;
        then
A18:    C3 \/ G0A is connected by A15,A16,CONNSP_1:1,17;
        G0A c= C3 \/ G0A by XBOOLE_1:7;
        then G0A=C3 \/ G0A by A10,A18;
        then C3 c= G0A by XBOOLE_1:7;
        hence C3=G0A or C3=G0B by A12,A17;
      end;
      suppose C3 /\ A2<>{};
        then
A19:    C3 meets G0B;
A20:    C3 is connected by A12;
A21:    G0B is connected by A11;
        then
A22:    C3 \/ G0B is connected by A19,A20,CONNSP_1:1,17;
        G0B c= C3 \/ G0B by XBOOLE_1:7;
        then G0B=C3 \/ G0B by A11,A22;
        then C3 c= G0B by XBOOLE_1:7;
        hence C3=G0A or C3=G0B by A12,A21;
      end;
    end;
    hence C3=G0A or C3=G0B;
  end;
  hence thesis by A2,A3,A4,A10,A11;
end;
