reserve X for non empty TopSpace,
  A,B for Subset of X;
reserve Y1,Y2 for non empty SubSpace of X;
reserve X1, X2 for non empty SubSpace of X;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X for non discrete non empty TopSpace;

theorem
  for Y1, Y2 being non empty TopSpace st Y2 = the TopStruct of Y1 holds
  Y1 is boundary SubSpace of X iff Y2 is boundary SubSpace of X
proof
  let X1,X2 be non empty TopSpace;
  assume
A1: X2 = the TopStruct of X1;
  thus X1 is boundary SubSpace of X implies X2 is boundary SubSpace of X
  proof
    assume
A2: X1 is boundary SubSpace of X;
    then reconsider Y2 = X2 as SubSpace of X by A1,TMAP_1:7;
    for A2 be Subset of X st A2 = the carrier of Y2 holds A2 is boundary
    by A1,A2,Def3;
    hence thesis by Def3;
  end;
  assume
A3: X2 is boundary SubSpace of X;
  then reconsider Y1 = X1 as SubSpace of X by A1,TMAP_1:7;
  for A1 be Subset of X st A1 = the carrier of Y1 holds A1 is boundary by A1,A3
,Def3;
  hence thesis by Def3;
end;
