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 Th9:
  fi is increasing & A c= B & B in dom fi implies fi.A c= fi.B
proof
  assume that
A1: for A,B st A in B & B in dom fi holds fi.A in fi.B and
A2: A c= B and
A3: B in dom fi;
  reconsider C = fi.B as Ordinal;
  now
    per cases;
    suppose
      A = B;
      hence thesis;
    end;
    suppose
      A <> B;
      then A c< B by A2;
      then A in B by ORDINAL1:11;
      then fi.A in C by A1,A3;
      hence thesis by ORDINAL1:def 2;
    end;
  end;
  hence thesis;
end;
