reserve n,m,k,k1,k2 for Nat;
reserve r,r1,r2,s,t,p for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve x,y for set;

theorem Th43:
  seq is bounded_below implies ((for m st n <= m holds r <= seq.m)
  iff r <= (inferior_realsequence seq).n)
proof
  assume
A1: seq is bounded_below;
  thus (for m st n<=m holds r <= seq.m) implies r <= (inferior_realsequence
  seq).n
  proof
    assume for m st n<=m holds r <= seq.m;
    then for k holds r <= seq.(n+k) by NAT_1:11;
    hence thesis by A1,Th42;
  end;
  assume
A2: r <= (inferior_realsequence seq).n;
  now
    let m;
    assume n<=m;
    then consider k being Nat such that
A3: m = n + k by NAT_1:10;
    thus r <= seq.m by A1,A2,A3,Th42;
  end;
  hence thesis;
end;
