theorem Th7:
  for A being Subset of TOP-REAL n, B being Subset of TOP-REAL n
  holds B is_inside_component_of A iff ex C being Subset of
  ((TOP-REAL n) | (A`))
  st C=B & C is a_component & C is bounded Subset of Euclid n
proof
  let A be Subset of TOP-REAL n, B be Subset of TOP-REAL n;
A1: B is_a_component_of A` iff ex C being Subset of (TOP-REAL n) | (A`) st C=B
  & C is a_component by CONNSP_1:def 6;
  thus B is_inside_component_of A implies ex C being Subset of ((TOP-REAL n) |
(A`)) st C=B & C is a_component & C is bounded Subset of Euclid n
  by Th5,A1;
  given C being Subset of ((TOP-REAL n) | (A`)) such that
A2: C=B & C is a_component & C is bounded Subset
  of Euclid n;
  B is bounded & B is_a_component_of A` by A2,Th5,CONNSP_1:def 6;
  hence thesis;
end;
