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 Th11:
  phi is increasing implies phi is one-to-one
proof
  assume
A1: for A,B st A in B & B in dom phi holds phi.A in phi.B;
  let x,y be object;
  assume that
A2: x in dom phi & y in dom phi and
A3: phi.x = phi.y;
  reconsider A = x, B = y as Ordinal by A2;
A4: A in B or A = B or B in A by ORDINAL1:14;
  not phi.A in phi.B by A3;
  hence thesis by A1,A2,A3,A4;
end;
