reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2, X3 for non empty SubSpace of X;
reserve X1, X2, X3 for non empty SubSpace of X;
reserve X for TopSpace;
reserve A1, A2 for Subset of X;

theorem Th35:
  A1 \/ A2 is closed & A1,A2 are_separated implies A1 is closed & A2 is closed
proof
  assume
A1: A1 \/ A2 is closed;
  then Cl A1 c= A1 \/ A2 by TOPS_1:5,XBOOLE_1:7;
  then
A2: Cl A1 \ A2 c= A1 by XBOOLE_1:43;
  assume
A3: A1,A2 are_separated;
  then Cl A1 misses A2 by CONNSP_1:def 1;
  then
A4: Cl A1 c= A1 by A2,XBOOLE_1:83;
  Cl A2 c= A1 \/ A2 by A1,TOPS_1:5,XBOOLE_1:7;
  then
A5: Cl A2 \ A1 c= A2 by XBOOLE_1:43;
  Cl A2 misses A1 by A3,CONNSP_1:def 1;
  then
A6: Cl A2 c= A2 by A5,XBOOLE_1:83;
  A1 c= Cl A1 & A2 c= Cl A2 by PRE_TOPC:18;
  hence thesis by A6,A4,XBOOLE_0:def 10;
end;
