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 positive Real, n be non trivial Nat holds
    (a+b)|^n > a|^n + b|^n
  proof
    let a,b be positive Real, n be non trivial Nat;
    reconsider m = n-1 as non zero Nat;
    reconsider c = max(a,b), d = min(a,b) as positive Real;
A1: c*c|^m = c|^(m+1) & d*d|^m = d|^(m+1) &
      (c+d)*(c+d)|^m = (c+d)|^(m+1) by NEWTON:6;
A2: c|^n = max(a|^n, b|^n) & d|^n = min(a|^n,b|^n) by MM1;
    c+d > c+0 & c+d > 0+d by XREAL_1:6; then
    (c+d)|^m > c|^m & (c+d)|^m > d|^m by NEWTON02:40; then
    c*(c+d)|^m > c*c|^m & d*(c+d)|^m > d*d|^m by XREAL_1:68; then
     c*(c+d)|^m > c|^(m+1) & d*(c+d)|^m > d|^(m+1) by NEWTON:6; then
A3: c*(c+d)|^m + d*(c+d)|^m > c|^(m+1)+ d|^(m+1) by XREAL_1:8;
    (a+b)|^(m+1) = c*(c+d)|^m + d*(c+d)|^m by A1,MinMax;
    hence thesis by A2,A3,MinMax;
  end;
