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 Th14:
  for phi being Ordinal-Sequence st phi is increasing holds C+^phi
  is increasing
proof
  let phi be Ordinal-Sequence such that
A1: phi is increasing;
  let A,B;
  set xi = C+^phi;
  assume that
A2: A in B and
A3: B in dom xi;
  reconsider A9 = phi.A, B9 = phi.B as Ordinal;
A4: dom xi = dom phi by ORDINAL3:def 1;
  then
A5: xi.B = C+^B9 by A3,ORDINAL3:def 1;
  A in dom xi by A2,A3,ORDINAL1:10;
  then
A6: xi.A = C+^A9 by A4,ORDINAL3:def 1;
  A9 in B9 by A1,A2,A3,A4;
  hence thesis by A6,A5,ORDINAL2:32;
end;
