theorem Th137:
  v is increasing implies
  for n,m st n in dom v & m in dom v & n <>  m holds v.n <> v.m
proof
  assume
A1: v is increasing;
  let n,m;
  assume that
A2: n in dom v & m in dom v and
A3: n<>m;
  now
    per cases by A3,XXREAL_0:1;
    suppose
      n<m;
      hence thesis by A1,A2,SEQM_3:def 1;
    end;
    suppose
      n>m;
      hence thesis by A1,A2,SEQM_3:def 1;
    end;
  end;
  hence thesis;
end;
