reserve m,n,i,i2,j for Nat,
  r,r1,r2,s,t for Real,
  x,y,z for object;
reserve p,p1,p2,p3,q,q1,q2,q3,q4 for Point of TOP-REAL n;
reserve u for Point of Euclid n;
reserve R for Subset of TOP-REAL n;
reserve P,Q for Subset of TOP-REAL n;
reserve D for non vertical non horizontal non empty compact Subset of TOP-REAL
  2;
reserve f for clockwise_oriented non constant standard
  special_circular_sequence;

theorem
  for A being Subset of TOP-REAL n, B being Subset of TOP-REAL n st B
  is_inside_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_inside_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
  C is bounded Subset of Euclid n by Th7;
  C is connected by A2,CONNSP_1:def 5;
  hence thesis by A1,CONNSP_1:23;
end;
