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;
reserve W for Universe;
reserve A1,B1 for Ordinal of W,
  phi for Ordinal-Sequence of W;

theorem Th35:
  dom fi in W & rng fi c= W implies sup fi in W
proof
  assume that
A1: dom fi in W and
A2: rng fi c= W;
  ex A st rng fi c= A by ORDINAL2:def 4;
  then for x st x in rng fi holds x is Ordinal;
  then reconsider B = union rng fi as epsilon-transitive epsilon-connected set
by ORDINAL1:23;
A3: rng fi = fi.:(dom fi) by RELAT_1:113;
A4: sup fi c= succ B
  proof
    let x be object;
    assume
A5: x in sup fi;
    then reconsider A = x as Ordinal;
    consider C such that
A6: C in rng fi and
A7: A c= C by A5,ORDINAL2:21;
    C c= union rng fi by A6,ZFMISC_1:74;
    then A c= B by A7;
    hence thesis by ORDINAL1:22;
  end;
  card dom fi in card W by A1,CLASSES2:1;
  then card rng fi in card W by A3,CARD_1:67,ORDINAL1:12;
  then rng fi in W by A2,CLASSES1:1;
  then union rng fi in W by CLASSES2:59;
  then succ B in W by CLASSES2:5;
  hence thesis by A4,CLASSES1:def 1;
end;
