reserve H,S for ZF-formula,
  x for Variable,
  X,Y for set,
  i for Element of NAT,
  e,u for set;
reserve M,M1,M2 for non empty set,
  f for Function,
  v1 for Function of VAR,M1,
  v2 for Function of VAR,M2,
  F,F1,F2 for Subset of WFF,
  W for Universe,
  a,b,c for Ordinal of W,
  A,B,C for Ordinal,
  L for DOMAIN-Sequence of W,
  va for Function of VAR,L.a,
  phi,xi for Ordinal-Sequence of W;

theorem Th16:
  for phi being Ordinal-Sequence st phi is increasing & phi is
  continuous holds C+^phi is continuous
proof
  let phi be Ordinal-Sequence such that
A1: phi is increasing;
  assume
A2: for A,B st A in dom phi & A <> 0 & A is limit_ordinal & B = phi.A
  holds B is_limes_of phi|A;
  let A,B;
  set xi = phi|A;
  reconsider A9 = phi.A as Ordinal;
  assume that
A3: A in dom (C+^phi) and
A4: A <> 0 and
A5: A is limit_ordinal and
A6: B = (C+^phi).A;
A7: dom phi = dom (C+^phi) by ORDINAL3:def 1;
  then
A8: B = C+^A9 by A3,A6,ORDINAL3:def 1;
  A9 is_limes_of xi by A2,A3,A4,A5,A7;
  then
A9: lim xi = A9 by ORDINAL2:def 10;
A10: dom xi = dom (C+^xi) & (C+^phi)|A = C+^xi by Th15,ORDINAL3:def 1;
A11: xi is increasing by A1,ORDINAL4:15;
  then
A12: C+^xi is increasing by Th14;
  A c= dom (C+^phi) by A3,ORDINAL1:def 2;
  then
A13: dom xi = A by A7,RELAT_1:62;
  then
A14: sup (C+^xi) = C+^sup xi by A4,ORDINAL3:43;
  sup xi = lim xi by A4,A5,A13,A11,ORDINAL4:8;
  hence thesis by A4,A5,A13,A10,A8,A14,A9,A12,ORDINAL4:8;
end;
