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_separated implies Y1,Y2 are_separated
proof
  let X1, X2 be non empty SubSpace of X;
  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_separated;
  then (X1 meet X0),(X2 meet X0) are_separated by A1,TSEP_1:70;
  hence thesis by A2,Th44;
end;
