reserve a,b,k,m,n,s for Nat;
reserve c,c1,c2,c3 for Complex;
reserve i,j,z for Integer;
reserve p for Prime;
reserve x for object;
reserve f,g for complex-valued FinSequence;
reserve a,b,x,y for Real;

theorem :: Problem 198 b
  for k,x,y being positive Nat holds
  a > 0 & x|^(2*k) - y|^(2*k) = a implies
  x < k-Root a & y < k-Root a
  proof
    let k,x,y be positive Nat such that
A1: a > 0 and
A2: x|^(2*k) - y|^(2*k) = a;
A3: 1 <= k by NAT_1:14;
    x|^(2*k) = x|^k|^2 & y|^(2*k) = y|^k|^2 by NEWTON:9;
    then
A4: x|^(2*k) - y|^(2*k) = (x|^k-y|^k) * (x|^k+y|^k) by NEWTON01:1;
    then x|^k-y|^k > 0 by A1,A2;
    then x|^k-y|^k >= 0+1 by NAT_1:13;
    then
A5: x|^k+y|^k <= 1*a by A2,A4,XREAL_1:97;
    x|^k+0 < x|^k+y|^k & y|^k+0 < x|^k+y|^k by XREAL_1:6;
    then x|^k < a & y|^k < a by A5,XXREAL_0:2;
    then k-Root(x|^k) < k-Root a & k-Root(y|^k) < k-Root a
    by NAT_1:14,PREPOWER:28;
    hence thesis by A3,PREPOWER:19;
  end;
