reserve a, r, s for Real;

theorem Th7:
  for S, T being TopSpace, A being Subset of S, B being Subset of T
st the TopStruct of S = the TopStruct of T & A = B & A is connected holds B is
  connected
proof
  let S, T be TopSpace, A be Subset of S, B be Subset of T such that
A1: the TopStruct of S = the TopStruct of T and
A2: A = B & A is connected;
  now
    let P, Q be Subset of T such that
A3: B = P \/ Q and
A4: P,Q are_separated;
    reconsider P1 = P, Q1 = Q as Subset of S by A1;
    P1,Q1 are_separated by A1,A4,Th5;
    then P1 = {}S or Q1 = {}S by A2,A3,CONNSP_1:15;
    hence P = {}T or Q = {}T;
  end;
  hence thesis by CONNSP_1:15;
end;
