reserve n,n1,n2,m for Nat,
  r,r1,r2,p,g1,g2,g for Real,
  seq,seq9,seq1 for Real_Sequence,
  y for set;

theorem Th17:
  seq is convergent & (for n holds 0<=seq.n) implies 0<=lim seq
proof
  assume that
A1: seq is convergent and
A2: for n holds 0<=seq.n and
A3: not 0<=(lim seq);
  0<-(lim seq) by A3;
  then consider n1 such that
A4: for m st n1<=m holds |.seq.m-lim seq.|<-(lim seq) by A1,Def6;
  |.seq.n1-lim seq.|<=-(lim seq) by A4;
  then seq.n1-(lim seq)<=-(lim seq) by ABSVALUE:5;
  then
A5: seq.n1-(lim seq)+(lim seq)<=-(lim seq)+(lim seq) by XREAL_1:6;
  now
    assume seq.n1=0;
    then |.seq.n1-lim seq.|=-(lim seq) by A3,ABSVALUE:def 1;
    hence contradiction by A4;
  end;
  hence contradiction by A2,A5;
end;
