reserve phi,fi,psi for Ordinal-Sequence,
  A,A1,B,C,D for Ordinal,
  f,g for Function,
  X for set,
  x,y,z for object;
reserve f1,f2 for Ordinal-Sequence;

theorem Th16:
  phi is increasing & dom phi is limit_ordinal implies sup phi is limit_ordinal
proof
  assume that
A1: phi is increasing and
A2: dom phi is limit_ordinal;
  now
    let A;
    assume A in sup phi;
    then consider B such that
A3: B in rng phi and
A4: A c= B by ORDINAL2:21;
    consider x being object such that
A5: x in dom phi and
A6: B = phi.x by A3,FUNCT_1:def 3;
    reconsider x as Ordinal by A5;
A7: succ x in dom phi by A2,A5,ORDINAL1:28;
    reconsider C = phi.succ x as Ordinal;
    x in succ x by ORDINAL1:6;
    then B in C by A1,A6,A7;
    then
A8: succ B c= C by ORDINAL1:21;
A9: succ A c= succ B by A4,ORDINAL2:1;
    C in rng phi by A7,FUNCT_1:def 3;
    then C in sup phi by ORDINAL2:19;
    then succ B in sup phi by A8,ORDINAL1:12;
    hence succ A in sup phi by A9,ORDINAL1:12;
  end;
  hence thesis by ORDINAL1:28;
end;
