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
  a>0 implies a|^(n+2) + a|^(n+2) <> b|^(n+2)
  proof
    assume
    A1: a>0;
    assume not thesis;
    then consider j,k,l such that
    A3: j|^(n+2) + k|^(n+2) = l|^(n+2) & j,k are_coprime
    & j,l are_coprime & k,l are_coprime
    & a = (a gcd a)*j & a = (a gcd a)*k & b = (a gcd a)*l by A1,Th41;
    k*a = 1*a & j*a = 1*a by NAT_D:32,A3; then
    k = 1 & j = 1 by A1,XCMPLX_1:5;
    hence contradiction by A3,Lm60;
  end;
