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 X1,Y1
constitute_a_decomposition & X2,Y2 constitute_a_decomposition holds Y1 union Y2
  = the TopStruct of X & Y1,Y2 are_weakly_separated implies X1,X2 are_separated
proof
  let X1, X2, Y1, Y2 be non empty SubSpace of X;
  assume
A1: X1,Y1 constitute_a_decomposition & X2,Y2 constitute_a_decomposition;
  assume Y1 union Y2 = the TopStruct of X;
  then
A2: X1 misses X2 by A1,Th32;
  assume Y1,Y2 are_weakly_separated;
  hence thesis by A1,A2,Th39;
end;
