reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2, X3 for non empty SubSpace of X;
reserve X1, X2, X3 for non empty SubSpace of X;
reserve X for TopSpace;
reserve A1, A2 for Subset of X;
reserve A1,A2 for Subset of X;
reserve X for TopSpace,
  A1, A2 for Subset of X;
reserve X for non empty TopSpace,
  A1, A2 for Subset of X;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;

theorem
  for Y, X1, X2 being non empty SubSpace of X st X1 meets Y & X2 meets Y
holds X1,X2 are_separated implies X1 meet Y,X2 meet Y are_separated & Y meet X1
  ,Y meet X2 are_separated
proof
  let Y, X1, X2 be non empty SubSpace of X such that
A1: X1 meets Y and
A2: X2 meets Y;
  reconsider A2 = the carrier of X2 as Subset of X by Th1;
  reconsider A1 = the carrier of X1 as Subset of X by Th1;
  reconsider C = the carrier of Y as Subset of X by Th1;
  assume X1,X2 are_separated;
  then
A3: A1,A2 are_separated;
  now
    let D1, D2 be Subset of X;
    assume D1 = the carrier of X1 meet Y & D2 = the carrier of X2 meet Y;
    then A1 /\ C = D1 & A2 /\ C = D2 by A1,A2,Def4;
    hence D1,D2 are_separated by A3,Th39;
  end;
  hence X1 meet Y,X2 meet Y are_separated;
  then X1 meet Y,Y meet X2 are_separated by A2,Th26;
  hence Y meet X1,Y meet X2 are_separated by A1,Th26;
end;
