reserve X for non empty TopSpace;
reserve Y for non empty TopStruct;
reserve x for Point of Y;
reserve Y for non empty TopStruct;
reserve X for non empty TopSpace;
reserve x,y for Point of X;
reserve A, B for Subset of X;
reserve P, Q for Subset of X;
reserve Y for non empty TopStruct;
reserve X for non empty TopSpace,
  Y0 for non empty SubSpace of X;

theorem
  for x, y being Point of Y holds y is Point of MaxADSspace(x) iff the
  TopStruct of MaxADSspace(y) = the TopStruct of MaxADSspace(x)
proof
  let x, y be Point of Y;
A1: the carrier of MaxADSspace(x) = MaxADSet(x) by Def17;
  thus y is Point of MaxADSspace(x) implies the TopStruct of MaxADSspace(y) =
  the TopStruct of MaxADSspace(x)
  proof
    assume y is Point of MaxADSspace(x);
    then MaxADSet(y) = MaxADSet(x) by A1,Th21;
    hence thesis by A1,Def17;
  end;
  assume
A2: the TopStruct of MaxADSspace(y) = the TopStruct of MaxADSspace(x);
  the carrier of MaxADSspace(y) = MaxADSet(y) by Def17;
  hence thesis by A2,Th21;
end;
