reserve GX for TopSpace;
reserve A, B, C for Subset of GX;
reserve TS for TopStruct;
reserve K, K1, L, L1 for Subset of TS;

theorem Th34:
  A is a_component & B is a_component implies
  A = B or A,B are_separated
proof
  assume that
A1: A is a_component and
A2: B is a_component;
  A <> B implies A,B are_separated
  proof
A3: B c= A \/ B by XBOOLE_1:7;
A4: A c= A \/ B by XBOOLE_1:7;
    assume
A5: A <> B;
    assume not A,B are_separated;
    then A \/ B is connected by A1,A2,Th17;
    then A = A \/ B by A1,A4;
    hence contradiction by A1,A2,A5,A3;
  end;
  hence thesis;
end;
