reserve
  a,b,c,d,e for Ordinal,
  m,n for Nat,
  f for Ordinal-Sequence,
  x for object;
reserve S,S1,S2 for Sequence;

theorem Th6:
  for f being non-decreasing Ordinal-Sequence
  st dom f is non empty holds Union f is_limes_of f
  proof
    let f be non-decreasing Ordinal-Sequence such that
A1: dom f is non empty;
    set a0 = the Element of dom f;
    per cases;
    case
A2:   Union f = 0;
      take a0; thus a0 in dom f by A1;
      let c; assume a0 c= c & c in dom f; then
      f.c in rng f by FUNCT_1:def 3;
      hence f.c = 0 by A2,ORDERS_1:6;
    end;
    case Union f <> 0;
      let b,c; assume
A3:   b in Union f & Union f in c; then
      consider x being object such that
A4:   x in dom f & b in f.x by CARD_5:2;
      reconsider x as Ordinal by A4;
      take x; thus x in dom f by A4;
      let E be Ordinal; assume
A5:   x c= E & E in dom f; then
      x = E or x c< E; then
      x = E or x in E by ORDINAL1:11; then
      f.x c= f.E by A5,Def2;
      hence b in f.E by A4;
      f.E in rng f by A5,FUNCT_1:def 3; then
      f.E c= Union f by ZFMISC_1:74;
      hence f.E in c by A3,ORDINAL1:12;
    end;
  end;
