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;
reserve X0 for non empty SubSpace of X;

theorem
  for X1, X2 being non empty SubSpace of X st X1 meets X0 & X2 meets X0
for Y1, Y2 being SubSpace of X0 st Y1 = X1 meet X0 & Y2 = X2 meet X0 holds X1,
  X2 are_weakly_separated implies Y1,Y2 are_weakly_separated
proof
  let X1, X2 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;
  assume
A1: X1 meets X0 & X2 meets X0;
  let Y1, Y2 be SubSpace of X0;
  assume
A2: Y1 = X1 meet X0 & Y2 = X2 meet X0;
  assume X1,X2 are_weakly_separated;
  then
A3: A1,A2 are_weakly_separated;
  now
    let C1, C2 be Subset of X0;
    assume C1 = the carrier of Y1 & C2 = the carrier of Y2;
    then
    C1 = (the carrier of X0) /\ A1 & C2 = (the carrier of X0) /\ A2 by A1,A2,
TSEP_1:def 4;
    hence C1,C2 are_weakly_separated by A3,Th28;
  end;
  hence thesis;
end;
