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

theorem
  for p be prime Nat holds
  a*b = p|^n implies ex k,l be Nat st
  a = p|^k & b= p|^l & k+l = n
  proof
    let p be prime Nat;
    assume
    A1: a*b = p|^n; then
    a divides p|^n; then
    consider k be Nat such that
    A2: a = p|^k & k <= n by GROUPP_1:2;
    b divides p|^n by A1; then
    consider l be Nat such that
    A3: b = p|^l & l <= n by GROUPP_1:2;
    p|^n = p|^(k+l) by NEWTON:8,A1,A2,A3; then
    n = k+l by PEPIN:30,Def0;
    hence thesis by A2,A3;
  end;
