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 Th37:
  for X1, X2, Y1, Y2 being SubSpace of X st X1,Y1
  constitute_a_decomposition & X2,Y2 constitute_a_decomposition holds X1,X2
  are_weakly_separated implies Y1,Y2 are_weakly_separated
proof
  let X1, X2, Y1, Y2 be SubSpace of X;
  assume
A1: for A1, B1 being Subset of X st A1 = the carrier of X1 & B1 = the
  carrier of Y1 holds A1,B1 constitute_a_decomposition;
  assume
A2: for A2, B2 being Subset of X st A2 = the carrier of X2 & B2 = the
  carrier of Y2 holds A2,B2 constitute_a_decomposition;
  assume
A3: for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the
  carrier of X2 holds A1,A2 are_weakly_separated;
  now
    reconsider A1 = the carrier of X1, A2 = the carrier of X2 as Subset of X
    by TSEP_1:1;
    let B1, B2 be Subset of X;
    assume B1 = the carrier of Y1 & B2 = the carrier of Y2;
    then
A4: A1,B1 constitute_a_decomposition & A2,B2 constitute_a_decomposition by A1
,A2;
    A1,A2 are_weakly_separated by A3;
    hence B1,B2 are_weakly_separated by A4,Th15;
  end;
  hence thesis;
end;
