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, a,b be Nat st a <> b holds
    p |-count (a + b) >= p |-count (a gcd b)
  proof
    let p be prime Nat, a,b be Nat such that
A1: a <> b;
A1a: a <> 0 or b <> 0 by A1; then
    consider k,l be Integer such that
A2: a = (a gcd b)*k & b = (a gcd b)*l & k,l are_coprime by INT_2:23;
    k >= 0 & l >= 0 by A1a,A2; then
    k in NAT & l in NAT by INT_1:3; then
    reconsider k,l as Nat;
A4: k is non zero or l is non zero by A2;
    p |-count (a+b) = p |-count ((k+l)*(a gcd b)) by A2
    .= (p |-count (k+l)) + (p|-count (a gcd b)) by A1a,A4,NAT_3:28;
    hence thesis by NAT_1:12;
  end;
