reserve n,m,k for Nat;
reserve r,r1 for Real;
reserve f,seq,seq1 for Real_Sequence;
reserve x,y for set;
reserve e1,e2 for ExtReal;
reserve Nseq for increasing sequence of NAT;

theorem
  seq is bounded_below & seq1 is non-decreasing implies seq+seq1 is
  bounded_below
proof
  assume that
A1: seq is bounded_below and
A2: seq1 is non-decreasing;
  consider r1 such that
A3: for n holds r1<seq.n by A1;
  take r=r1+seq1.0;
  let n be Nat;
  seq1.0<=seq1.n by A2,Th11;
  then r1+seq1.0<seq.n+seq1.n by A3,XREAL_1:8;
  hence r<(seq+seq1).n by SEQ_1:7;
end;
