let n be even Nat; :: thesis: for x being Element of F_Real holds x |^ n >= 0. F_Real
let x be Element of F_Real; :: thesis: x |^ n >= 0. F_Real
defpred S₁[ Nat] means x |^ (2 * $1) >= 0. F_Real;
x |^ 0 = 1_ F_Real by BINOM:8;
then IA: S₁[ 0 ] ;
XX1: for x being Element of F_Real holds x |^ 2 >= 0. F_Real

proof

let x be Element of F_Real; :: thesis: x |^ 2 >= 0. F_Real

per cases ( x >= 0 or x <= 0 ) ;

suppose x >= 0 ; :: thesis: x |^ 2 >= 0. F_Real

then x * x >= 0 * 0 ;
hence x |^ 2 >= 0. F_Real by prl4; :: thesis: verum

end;

suppose x <= 0 ; :: thesis: x |^ 2 >= 0. F_Real

then x * x >= 0 * 0 ;
hence x |^ 2 >= 0. F_Real by prl4; :: thesis: verum

end;

end;

end;

IS: now :: thesis: for k being Nat st S₁[k] holds
S₁[k + 1]

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume AS: S₁[k] ; :: thesis: S₁[k + 1]
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 S₁[k + 1] by H0, AS; :: thesis: verum

end;

I: for k being Nat holds S₁[k] from NAT_1:sch 2(IA, IS);
ex k being Nat st n = 2 * k by ABIAN:def 2;
hence x |^ n >= 0. F_Real by I; :: thesis: verum