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 a,b be non zero Nat, p be prime Nat holds
  a|^n + b|^n = p|^n implies a,b are_coprime
  proof
    let a,b be non zero Nat, p be prime Nat;
    A1: a|^n+b|^n > a|^n+0 by XREAL_1:6;
    assume
    A2: a|^n + b|^n = p|^n; then
    reconsider n as non zero Nat;
    a|^n < p|^n by A1,A2; then
    a < p by NEWTON02:40; then
    a,p are_coprime by NAT_D:7,NAT_6:6; then
    a|^n, p|^n are_coprime by WSIERP_1:11; then
    a|^n, b|^n are_coprime by A2, NEWTON02:45; then
    a, b|^n are_coprime by NEWTON02:73;
    hence thesis by NEWTON02:73;
  end;
