reserve a,b,c,d,x,j,k,l,m,n for Nat,
  p,q,t,z,u,v for Integer,
  a1,b1,c1,d1 for Complex;

theorem Th32:
  a-b divides a|^n-b|^n
  proof
    defpred P[Nat] means a-b divides a|^$1 -b|^$1;
    A1: P[0] by Lm18b;
    A2: P[x] implies P[x+1]
    proof
      x = 0 or x > 0;
      hence thesis by Lm18f;
    end;
    for m holds P[m] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
