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;
