reserve X for non empty 1-sorted;
reserve A, A1, A2, B1, B2 for Subset of X;
reserve X for non empty TopSpace;
reserve A, A1, A2, B1, B2 for Subset of X;
reserve X for non empty 1-sorted;
reserve A, A1, A2, B1, B2 for Subset of X;
reserve X for non empty TopSpace,
  A1, A2 for Subset of X;
reserve X0 for non empty SubSpace of X,
  B1, B2 for Subset of X0;

theorem Th26:
  B1 = (the carrier of X0) /\ A1 & B2 = (the carrier of X0) /\ A2
  implies (A1,A2 are_separated implies B1,B2 are_separated)
proof
  assume
A1: B1 = (the carrier of X0) /\ A1;
  then reconsider C1 = B1 as Subset of X;
  assume
A2: B2 = (the carrier of X0) /\ A2;
  then
A3: B2 c= A2 by XBOOLE_1:17;
  reconsider C2 = B2 as Subset of X by A2;
  assume
A4: A1,A2 are_separated;
  B1 c= A1 by A1,XBOOLE_1:17;
  then C1,C2 are_separated by A3,A4,CONNSP_1:7;
  hence thesis by Th25;
end;
