reserve x, y, z, w for Real;
reserve n for Element of NAT;

theorem
  (cosh(x)+sinh(x)) |^ n = cosh(n*x)+sinh(n*x) & (cosh(x)-sinh(x)) |^ n
  = cosh(n*x)-sinh(n*x)
proof
A1: (cosh(x) + sinh(x)) |^ n = (cosh.x+sinh(x)) |^ n by SIN_COS2:def 4
    .= (cosh.x+sinh.x) |^ n by SIN_COS2:def 2
    .= cosh.(n*x)+sinh.(n*x) by SIN_COS2:29
    .= cosh(n*x)+sinh.(n*x) by SIN_COS2:def 4
    .= cosh(n*x)+sinh(n*x) by SIN_COS2:def 2;
  (cosh(x) - sinh(x)) |^ n = (cosh(x)+-sinh(x)) |^ n
    .= (cosh(x)+sinh(-x)) |^ n by Lm8
    .= (cosh.x+sinh(-x)) |^ n by SIN_COS2:def 4
    .= (cosh.x+sinh.-x) |^ n by SIN_COS2:def 2
    .= (cosh.-x+sinh.-x) |^ n by SIN_COS2:19
    .= cosh.(n*-x)+sinh.(-n*x) by SIN_COS2:29
    .= cosh.(n*x)+sinh.(-n*x) by SIN_COS2:19
    .= cosh.(n*x)+-sinh.(n*x) by SIN_COS2:19
    .= cosh.(n*x)-sinh.(n*x)
    .= cosh(n*x)-sinh.(n*x) by SIN_COS2:def 4
    .= cosh(n*x)-sinh(n*x) by SIN_COS2:def 2;
  hence thesis by A1;
end;
