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 Th28:
  B1 = (the carrier of X0) /\ A1 & B2 = (the carrier of X0) /\ A2
  implies (A1,A2 are_weakly_separated implies B1,B2 are_weakly_separated)
proof
  assume B1 = (the carrier of X0) /\ A1 & B2 = (the carrier of X0) /\ A2;
  then
A1: B1 \ B2 = (the carrier of X0) /\ (A1 \ A2) & B2 \ B1 = (the carrier of
  X0) /\ (A2 \ A1) by XBOOLE_1:50;
  assume A1 \ A2,A2 \ A1 are_separated;
  then B1 \ B2,B2 \ B1 are_separated by A1,Th26;
  hence thesis;
end;
