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
  (for A st A in dom fi holds fi.A = C+^A) implies fi is increasing
proof
  assume
A1: for A st A in dom fi holds fi.A = C+^A;
  let A,B;
  assume that
A2: A in B and
A3: B in dom fi;
A4: fi.B = C+^B by A1,A3;
  fi.A = C+^A by A1,A2,A3,ORDINAL1:10;
  hence thesis by A2,A4,ORDINAL2:32;
end;
