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 Th124:
  for X,Y being Subset of REAL st
  X <> {} & X is bounded_below & Y <> {} & Y is bounded_below holds
  lower_bound (X ++ Y) = lower_bound X + lower_bound Y
proof
  let X,Y be Subset of REAL such that
A1: X <> {} and
A2: X is bounded_below and
A3: Y <> {} and
A4: Y is bounded_below;
A5: now
    let r9 be Real;
    assume 0<r9;
    then
A6: r9/2 > 0;
    then consider r1 be Real such that
A7: r1 in X and
A8: r1<lower_bound X+r9/2 by A1,A2,Def2;
    consider r2 be Real such that
A9: r2 in Y and
A10: r2<lower_bound Y+r9/2 by A3,A4,A6,Def2;
     reconsider r = r1 + r2 as Real;
    take r;
    thus r in X++Y by A7,A9,MEMBER_1:46;
    lower_bound X + r9/2 + (lower_bound Y + r9/2) = lower_bound X +
    lower_bound Y + r9;
    hence r<lower_bound X + lower_bound Y+r9 by A8,A10,XREAL_1:8;
  end;
A11: now
    let r be Real;
    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 r11,r22 being Complex such that
A12: r = r11 + r22 and
A13: r11 in X & r22 in Y;
    reconsider r1 = r11, r2 = r22 as Real by A13;
    r1 >= lower_bound X & r2 >= lower_bound Y by A2,A4,A13,Def2;
    hence lower_bound X + lower_bound Y<=r by A12,XREAL_1:7;
  end;
  X ++ Y <> {} & X ++ Y is bounded_below by A1,A2,A3,A4,Th123;
  hence thesis by A11,A5,Def2;
end;
