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 Th71:
  for Y1, Y2 being SubSpace of X st Y1 is SubSpace of X1 & Y2 is
  SubSpace of X2 holds X1,X2 are_separated implies Y1,Y2 are_separated
proof
  reconsider A2 = the carrier of X2 as Subset of X by Th1;
  reconsider A1 = the carrier of X1 as Subset of X by Th1;
  let Y1, Y2 be SubSpace of X such that
A1: Y1 is SubSpace of X1 & Y2 is SubSpace of X2;
  assume
A2: X1,X2 are_separated;
  now
    let B1, B2 be Subset of X;
    assume B1 = the carrier of Y1 & B2 = the carrier of Y2;
    then
A3: B1 c= A1 & B2 c= A2 by A1,Th4;
    A1,A2 are_separated by A2;
    hence B1,B2 are_separated by A3,CONNSP_1:7;
  end;
  hence thesis;
end;
