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 Th12:
  f1 is increasing implies f2*f1 is Ordinal-Sequence
proof
A1: dom(f2*f1) = f1"dom f2 by RELAT_1:147;
  assume f1 is increasing;
  then dom(f2*f1) is Ordinal by A1,Th11;
  then reconsider f = f2*f1 as Sequence by ORDINAL1:def 7;
  consider A such that
A2: rng f2 c= A by ORDINAL2:def 4;
  rng f c= rng f2 by RELAT_1:26;
  then rng f c= A by A2;
  hence thesis by ORDINAL2:def 4;
end;
