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 Th8:
  dom fi <> {} & dom fi is limit_ordinal & fi is increasing implies
  sup fi is_limes_of fi & lim fi = sup fi
proof
  assume that
A1: dom fi <> {} & dom fi is limit_ordinal and
A2: A in B & B in dom fi implies fi.A in fi.B;
  reconsider x = fi.{} as Ordinal;
  {} in dom fi by A1,ORDINAL3:8;
  then
A3: x in rng fi by FUNCT_1:def 3;
  thus sup fi is_limes_of fi
  proof
    per cases;
    case
      sup fi = 0;
      hence thesis by A3,ORDINAL2:19;
    end;
    case
      sup fi <> 0;
      let A,B;
      assume that
A4:   A in sup fi and
A5:   sup fi in B;
      consider C such that
A6:   C in rng fi and
A7:   A c= C by A4,ORDINAL2:21;
      consider x being object such that
A8:   x in dom fi and
A9:   C = fi.x by A6,FUNCT_1:def 3;
      reconsider x as Ordinal by A8;
      take M = succ x;
      thus M in dom fi by A1,A8,ORDINAL1:28;
      let D;
      assume that
A10:  M c= D and
A11:  D in dom fi;
      reconsider E = fi.D as Ordinal;
      x in M by ORDINAL1:6;
      then C in E by A2,A9,A10,A11;
      hence A in fi.D by A7,ORDINAL1:12;
      fi.D in rng fi by A11,FUNCT_1:def 3;
      then E in sup fi by ORDINAL2:19;
      hence thesis by A5,ORDINAL1:10;
    end;
  end;
  hence thesis by ORDINAL2:def 10;
end;
