reserve n for Nat;

theorem XX:
for n being even Nat,
    x being Element of F_Real holds x|^n >= 0.F_Real
proof
let n be even Nat, x be Element of F_Real;
defpred P[Nat] means x|^(2*($1)) >= 0.F_Real;
x|^0 = 1_F_Real by BINOM:8;
then IA: P[0];
XX1: for x being Element of F_Real holds x|^2 >= 0.F_Real
proof
let x be Element of F_Real;
per cases;
suppose x >= 0;
  then x * x >= 0 * 0;
  hence thesis by prl4;
  end;
suppose x <= 0;
  then x * x >= 0 * 0;
  hence thesis by prl4;
  end;
end;
IS: now let k be Nat;
    assume AS: P[k];
    H0: x|^(2*(k+1)) = x|^(2*k+2) .= x|^(2*k) * x|^2 by BINOM:10;
    x|^2 >= 0.F_Real by XX1;
    hence P[k+1] by H0,AS;
    end;
I: for k being Nat holds P[k] from NAT_1:sch 2(IA,IS);
ex k being Nat st n = 2 * k by ABIAN:def 2;
hence thesis by I;
end;
