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 Th87: :: Problem 198 a
  for n,x,y being positive Nat holds
  a > 0 & 2 <= n & x|^n - y|^n = a implies
  x < (n-1) -Root a & y < (n-1) -Root a
  proof
    let n,x,y be positive Nat such that
A1: a > 0 and
A2: 2 <= n and
A3: x|^n - y|^n = a;
    set F = (x,y)Subnomial(n-1);
A4: 2-1 <= n-1 by A2,XREAL_1:7;
A5: a = (x-y) * Sum F by A3,NEWTON04:51;
    then
A6: x-y > 0 by A1;
    then reconsider w = x-y as Element of NAT by INT_1:3;
    w >= 0+1 by A6,NAT_1:13;
    then
A7: Sum F <= 1*a by A5,XREAL_1:97;
    x|^(n-1) + y|^(n-1) <= Sum F by A4,NEWTON04:84;
    then
A8: x|^(n-1) + y|^(n-1) <= a by A7,XXREAL_0:2;
    x|^(n-1) + 0 < x|^(n-1) + y|^(n-1) & y|^(n-1) + 0 < x|^(n-1) + y|^(n-1)
    by XREAL_1:6;
    then x|^(n-1) < a & y|^(n-1) < a by A8,XXREAL_0:2;
    then
    (n-1) -Root (x|^(n-1)) < (n-1) -Root a &
    (n-1) -Root (y|^(n-1)) < (n-1) -Root a by A4,PREPOWER:28;
    hence thesis by A4,PREPOWER:19;
  end;
