reserve n, k, r, m, i, j for Nat;

theorem Th37:
  (tau to_power r) ^2 - 2 * ((-1) to_power r) + (tau to_power (-r)
  ) ^2 = ((tau to_power r) - (tau_bar to_power r)) ^2
proof
  (-1) / tau < 0 by Th33;
  then -1 / tau < 0 by XCMPLX_1:187;
  then
A1: 1 / tau > -(0 qua Nat);
  ((tau to_power r) - (tau_bar to_power r)) ^2 = (tau to_power r) ^2 - 2 *
(tau to_power r) * ((-1 / tau) to_power r) + ((-1 / tau) to_power r) ^2 by Th36
    .= (tau to_power r) ^2 - 2 * (tau to_power r) * (((-1) * (1 / tau)) #Z r
  ) + ((-1 / tau) to_power r) ^2 by POWER:def 2
    .= (tau to_power r) ^2 - 2 * (tau to_power r) * (((-1) #Z r) * ((1 / tau
  ) #Z r)) + ((-1 / tau) to_power r) ^2 by PREPOWER:40
    .= (tau to_power r) ^2 - 2 * (tau to_power r) * (((1 / tau) |^ r) * ((-1
  ) #Z r)) + ((-1 / tau) to_power r) ^2 by PREPOWER:36
    .= (tau to_power r) ^2 - 2 * (tau |^ r) * (((1 / tau) |^ r) * ((-1) #Z r
  )) + ((-1 / tau) to_power r) ^2 by POWER:41
    .= (tau to_power r) ^2 - 2 * ((tau |^ r) * ((1 / tau) |^ r)) * ((-1) #Z
  r) + ((-1 / tau) to_power r) ^2
    .= (tau to_power r) ^2 - 2 * ((tau * (1 / tau))|^ r) * ((-1) #Z r) + ((-
  1 / tau) to_power r) ^2 by NEWTON:7
    .= (tau to_power r) ^2 - 2 * (1 |^ r) * ((-1) #Z r) + ((-1 / tau)
  to_power r) ^2 by Th33,XCMPLX_1:106
    .= (tau to_power r) ^2 - 2 * 1 * ((-1) #Z r) + ((-1 / tau) to_power r)
  ^2
    .= (tau to_power r) ^2 - 2 * ((-1) to_power r) + ((-1 / tau) to_power r)
  ^2 by POWER:def 2
    .= (tau to_power r) ^2 - 2 * ((-1) to_power r) + ((-1 / tau) #Z r) ^2 by
POWER:def 2
    .= (tau to_power r) ^2 - 2 * ((-1) to_power r) + (((-1 / tau) * (-1 /
  tau)) #Z r) by PREPOWER:40
    .= (tau to_power r) ^2 - 2 * ((-1) to_power r) + (((-1 / tau) ^2) |^ r)
  by PREPOWER:36
    .= (tau to_power r) ^2 - 2 * ((-1) to_power r) + (((1 / tau) ^2)
  to_power r) by POWER:41
    .= (tau to_power r) ^2 - 2 * ((-1) to_power r) + ((1 / tau) to_power r)
  ^2 by A1,POWER:30
    .= (tau to_power r) ^2 - 2 * ((-1) to_power r) + (tau to_power (-r)) ^2
  by Th33,POWER:32;
  hence thesis;
end;
