reserve a,b,c,d for positive Real,
  m,u,w,x,y,z for Real,
  n,k for Nat,
  s,s1 for Real_Sequence;

theorem
  a^2+b^2=c^2 & n>=3 implies a|^(n+2)+b|^(n+2)<c|^(n+2)
proof
  assume that
A1: a^2+b^2=c^2 and
A2: n>=3;
A3: n>=1 by A2,XXREAL_0:2;
  b^2>0;
  then
A4: b|^2>0 by NEWTON:81;
  c^2-b^2+b^2>0+b^2 by A1,XREAL_1:8;
  then sqrt(c^2)>sqrt(b^2) by SQUARE_1:27;
  then c>sqrt(b^2) by SQUARE_1:22;
  then c>b by SQUARE_1:22;
  then c|^n>b|^n by A3,PREPOWER:10;
  then c|^n*b|^2>b|^n*b|^2 by A4,XREAL_1:68;
  then
A5: c|^n*b|^2>b|^(n+2) by NEWTON:8;
  a^2>0;
  then
A6: a|^2>0 by NEWTON:81;
  c^2-a^2+a^2>0+a^2 by A1,XREAL_1:8;
  then sqrt(c^2)>sqrt(a^2) by SQUARE_1:27;
  then c>sqrt(a^2) by SQUARE_1:22;
  then c>a by SQUARE_1:22;
  then c|^n>a|^n by A3,PREPOWER:10;
  then c|^n*a|^2>a|^n*a|^2 by A6,XREAL_1:68;
  then c|^n*a|^2>a|^(n+2) by NEWTON:8;
  then c|^n*a|^2+c|^n*b|^2>a|^(n+2)+b|^(n+2) by A5,XREAL_1:8;
  then c|^n*(a|^2+b|^2)>a|^(n+2)+b|^(n+2);
  then c|^n*(a^2+b|^2)>a|^(n+2)+b|^(n+2) by NEWTON:81;
  then c|^n*c^2>a|^(n+2)+b|^(n+2) by A1,NEWTON:81;
  then c|^n*c|^2>a|^(n+2)+b|^(n+2) by NEWTON:81;
  hence thesis by NEWTON:8;
end;
