reserve n,k,k1,m,m1,n1,n2,l for Nat;
reserve r,r1,r2,p,p1,g,g1,g2,s,s1,s2,t for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve Nseq for increasing sequence of NAT;
reserve x for set;
reserve X,Y for Subset of REAL;
reserve k,n for Nat,
  r,r9,r1,r2 for Real,
  c,c9,c1,c2,c3 for Element of COMPLEX;
reserve z,z1,z2 for FinSequence of COMPLEX;
reserve x,z,z1,z2,z3 for Element of COMPLEX n,
  A,B for Subset of COMPLEX n;

theorem Th123:
  for X,Y being Subset of REAL holds X is bounded_below &
    Y is bounded_below implies X++Y is bounded_below
proof
  let X,Y be Subset of REAL;
  assume X is bounded_below;
  then consider r1 be Real such that
A1: r1 is LowerBound of X;
A2: for r be Real st r in X holds r1<=r by A1,XXREAL_2:def 2;
  assume Y is bounded_below;
  then consider r2 be Real such that
A3: r2 is LowerBound of Y;
A4: for r be Real st r in Y holds r2<=r by A3,XXREAL_2:def 2;
  take r1 + r2;
  let r be ExtReal;
  assume r in X++Y; then
  r in {r22+r12 where r22,r12 is Complex
     : r22 in X & r12 in Y} by MEMBER_1:def 6;
  then consider r22,r12 being Complex such that
A5: r = r22 + r12 and
A6: r22 in X and
A7: r12 in Y;
    reconsider r9 = r22, r99 = r12 as Real by A6,A7;
A8: r2 <= r99 by A4,A7;
  r1 <= r9 by A2,A6;
  hence thesis by A5,A8,XREAL_1:7;
end;
