
theorem Th7:
  for T being non empty TopSpace, A being non empty Subset of T, B1
,B2,C1,C2 being Subset of T st B1 is_a_component_of A & B2 is_a_component_of A
& C1 is_a_component_of A & C2 is_a_component_of A & B1 \/ B2 = A & C1 \/ C2 = A
  holds { B1,B2 } = { C1,C2 }
proof
  let T be non empty TopSpace, A be non empty Subset of T, B1,B2,C1,C2 be
  Subset of T such that
A1: B1 is_a_component_of A and
A2: B2 is_a_component_of A and
A3: C1 is_a_component_of A and
A4: C2 is_a_component_of A and
A5: B1 \/ B2 = A and
A6: C1 \/ C2 = A;
  now
    let x be object;
    x = B1 or x = B2 iff x = C1 or x = C2 by A1,A2,A3,A4,A5,A6,Th6;
    hence x in { B1,B2 } iff x = C1 or x = C2 by TARSKI:def 2;
  end;
  hence thesis by TARSKI:def 2;
end;
