theorem Th100:
  for A being Subset of TOP-REAL n, B being Subset of TOP-REAL n
  st B is_outside_component_of A holds B is connected
proof
  let A be Subset of TOP-REAL n, B be Subset of TOP-REAL n;
  assume B is_outside_component_of A;
  then consider C being Subset of (TOP-REAL n) | A` such that
A1: C = B and
A2: C is a_component and
  not C is bounded Subset of Euclid n by Th8;
  C is connected by A2,CONNSP_1:def 5;
  hence thesis by A1,CONNSP_1:23;
end;
