reserve k,n,m for Nat,
  A,B,C for Ordinal,
  X for set,
  x,y,z for object;
reserve f,g,h,fx for Function,
  K,M,N for Cardinal,
  phi,psi for
  Ordinal-Sequence;

theorem
  rng phi = rng psi & phi is increasing & psi is increasing implies phi
  = psi
proof
  assume
A1: rng phi = rng psi & phi is increasing & psi is increasing;
A2: dom phi = dom psi
  proof
    thus dom phi c= dom psi
    by A1,Lm3;
    let x be Ordinal;
    assume x in dom psi;
    hence thesis by A1,Lm3;
  end;
  for x being object st x in dom phi holds phi.x = psi.x by A1,Lm3;
  hence thesis by A2;
end;
