theorem
  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 such that
A2: n in dom v & m in dom v and
A3: n<=m;
  now
    per cases;
    suppose
      n=m;
      hence thesis;
    end;
    suppose
      n<>m;
      then n<m by A3,XXREAL_0:1;
      hence thesis by A1,A2,SEQM_3:def 1;
    end;
  end;
  hence thesis;
end;
