reserve a, b for Real,
  r for Real,
  rr for Real,
  i, j, n for Nat,
  M for non empty MetrSpace,
  p, q, s for Point of TOP-REAL 2,
  e for Point of Euclid 2,
  w for Point of Euclid n,
  z for Point of M,
  A, B for Subset of TOP-REAL n,
  P for Subset of TOP-REAL 2,
  D for non empty Subset of TOP-REAL 2;
reserve a, b for Real;
reserve a, b for Real;
reserve r for Real;

theorem Th72:
  P is bounded closed implies P is compact
proof
  assume
A1: P is bounded closed;
  then reconsider C = P as bounded Subset of Euclid 2 by JORDAN2C:11;
  consider r being Real, e such that
  0 < r and
A2: C c= Ball(e,r) by METRIC_6:def 3;
  reconsider p = e as Point of TOP-REAL 2 by TOPREAL3:8;
A3: ].p`2-r,p`2+r.[ c= [.p`2-r,p`2+r .] by XXREAL_1:25;
A4: Ball(e,r) c= product ((1,2)-->(].p`1-r,p`1+r.[,].p`2-r,p`2+r.[)) by Th40;
  ].p`1-r,p`1+r.[ c= [.p`1-r,p`1+r.] by XXREAL_1:25;
  then
  product ((1,2)-->(].p`1-r,p`1+r.[,].p`2-r,p`2+r.[)) c= product ((1,2)-->
  ([.p`1-r,p`1+r.],[.p`2-r,p`2+r.])) by A3,Th19;
  then
A5: Ball(e,r) c= product ((1,2)-->([.p`1-r,p`1+r.],[.p`2-r,p`2+r.])) by A4;
  product ((1,2)-->([.p`1-r,p`1+r.],[.p`2-r,p`2+r.])) is compact Subset of
  TOP-REAL 2 by Th71;
  hence thesis by A1,A2,A5,COMPTS_1:9,XBOOLE_1:1;
end;
