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 A, B being non empty Subset of Y holds A is Subset of MaxADSspace(
  B) implies MaxADSspace(A) is SubSpace of MaxADSspace(B)
proof
  let A, B be non empty Subset of Y;
  assume
A1: A is Subset of MaxADSspace(B);
A2: the carrier of MaxADSspace(B) = MaxADSet(B) by Def18;
  the carrier of MaxADSspace(A) = MaxADSet(A) by Def18;
  hence thesis by A2,A1,Lm2,Th34;
end;
