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;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;
reserve X for non empty TopSpace;

theorem
  for X1, X2, Y1, Y2 being non empty SubSpace of X st Y1 is SubSpace of
  X1 & Y2 is SubSpace of X2 & Y1 meets Y2 & Y1 meet Y2 = X1 meet X2 holds X1,X2
  are_weakly_separated implies Y1,Y2 are_weakly_separated
proof
  let X1, X2, Y1, Y2 be non empty SubSpace of X;
  reconsider A1 = the carrier of X1, A2 = the carrier of X2 as Subset of X by
TSEP_1:1;
  reconsider C1 = the carrier of Y1, C2 = the carrier of Y2 as Subset of X by
TSEP_1:1;
  assume Y1 is SubSpace of X1 & Y2 is SubSpace of X2;
  then
A1: C1 c= A1 & C2 c= A2 by TSEP_1:4;
  assume
A2: Y1 meets Y2;
  assume
A3: Y1 meet Y2 = X1 meet X2;
  assume X1,X2 are_weakly_separated;
  then
A4: A1,A2 are_weakly_separated;
  now
    let C1, C2 be Subset of X;
    assume
A5: C1 = the carrier of Y1 & C2 = the carrier of Y2;
    then C1 meets C2 by A2;
    then C1 /\ C2 <> {};
    then A1 /\ A2 <> {} by A1,A5,XBOOLE_1:3,27;
    then A1 meets A2;
    then X1 meets X2;
    then A1 /\ A2 = the carrier of X1 meet X2 by TSEP_1:def 4;
    then A1 /\ A2 = C1 /\ C2 by A2,A3,A5,TSEP_1:def 4;
    hence C1,C2 are_weakly_separated by A1,A4,A5,Th24;
  end;
  hence thesis;
end;
