reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;

theorem Th15:
  t divides z implies t|^n divides z|^n
  proof
    assume
    A1: t divides z;
    |.t.| in NAT & |.z.| in NAT by INT_1:3; then
    consider a,b such that
    A2: a = |.t.| & b = |.z.|;
    A3: |.t.||^n = |.t|^n.| & |.z.||^n = |.z|^n.| by TAYLOR_2:1;
    (a gcd b)|^n = |.t.||^n gcd |.z.||^n by A2,Th7
    .= t|^n gcd z|^n by A3,INT_2:34; then
    t|^n gcd z|^n = |.a.||^n by A1,A2,INT_2:16,Th3
    .= |.t|^n.| by A2,TAYLOR_2:1;
    hence thesis by Th3;
  end;
