reserve x, y, z for set,
  T for TopStruct,
  A for SubSpace of T,
  P, Q for Subset of T;
reserve TS for TopSpace;
reserve PS, QS for Subset of TS;
reserve S for non empty TopStruct;
reserve f for Function of T,S;
reserve SS for non empty TopSpace;
reserve f for Function of TS,SS;

theorem Th19:
  for X being TopStruct for Y being SubSpace of X, A being Subset
  of X, B being Subset of Y st A = B holds A is compact iff B is compact
proof
  let X be TopStruct;
  let Y be SubSpace of X, A be Subset of X, B be Subset of Y such that
A1: A = B;
A2: B c= [#] Y;
  hence A is compact implies B is compact by A1,Th2;
  assume B is compact;
  then for P being Subset of Y st P = A holds P is compact by A1;
  hence thesis by A1,A2,Th2;
end;
