reserve n,m,k,k1,k2 for Nat;
reserve X for non empty Subset of ExtREAL;
reserve Y for non empty Subset of REAL;

theorem Th3:
  X = Y & Y is bounded_below implies X is bounded_below & inf X = lower_bound Y
proof
  assume that
A1: X=Y and
A2: Y is bounded_below;
A3: for s be Real st s in Y holds inf X <= s by A1,XXREAL_2:3;
  not +infty in X by A1;
  then
A4: X <> {+infty} by TARSKI:def 1;
  for r be ExtReal st r in X
   holds lower_bound Y <= r by A1,A2,SEQ_4:def 2;
  then
A5: lower_bound Y is LowerBound of X by XXREAL_2:def 2;
  hence X is bounded_below by XXREAL_2:def 9;
  then inf X in REAL by A4,XXREAL_2:58;
  then
A6: inf X <= lower_bound Y by A3,SEQ_4:43;
  lower_bound Y <= inf X by A5,XXREAL_2:def 4;
  hence thesis by A6,XXREAL_0:1;
end;
