reserve k,j,n for Nat,
  r for Real;
reserve x,x1,x2,y for Element of REAL n;
reserve f for real-valued FinSequence;
reserve p,p1,p2,p3 for Point of TOP-REAL n,
  x,x1,x2,y,y1,y2 for Real;
reserve p,p1,p2 for Point of TOP-REAL 2;

theorem
  for P being Subset of TOP-REAL n, Q being non empty Subset of Euclid n
  holds P = Q implies (TOP-REAL n) |P = TopSpaceMetr((Euclid n) |Q)
proof
  let P be Subset of (TOP-REAL n), Q be non empty Subset of Euclid n;
  set M = TopSpaceMetr((Euclid n) |Q);
  the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by Def8;
  then M is SubSpace of the TopStruct of TOP-REAL n by TOPMETR:13;
  then reconsider M = TopSpaceMetr((Euclid n) |Q) as SubSpace of TOP-REAL n by
PRE_TOPC:29;
  assume P = Q;
  then [#](M) = P by TOPMETR:def 2;
  hence thesis by PRE_TOPC:def 5;
end;
