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 union Y2 = X1 union X2 holds Y1,Y2
  are_weakly_separated implies X1,X2 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 union Y2 = X1 union X2;
  assume Y1,Y2 are_weakly_separated;
  then
A3: C1,C2 are_weakly_separated;
  now
    let A1, A2 be Subset of X;
    assume
A4: A1 = the carrier of X1 & A2 = the carrier of X2;
    then A1 \/ A2 = the carrier of X1 union X2 by TSEP_1:def 2;
    then A1 \/ A2 = C1 \/ C2 by A2,TSEP_1:def 2;
    hence A1,A2 are_weakly_separated by A1,A3,A4,Th22;
  end;
  hence thesis;
end;
