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 S_{1}[ Nat] means x |^ (2 * $1) >= 0. F_Real;

x |^ 0 = 1_ F_Real by BINOM:8;

then IA: S_{1}[ 0 ]
;

XX1: for x being Element of F_Real holds x |^ 2 >= 0. F_Real_{1}[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

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

defpred S

x |^ 0 = 1_ F_Real by BINOM:8;

then IA: S

XX1: for x being Element of F_Real holds x |^ 2 >= 0. F_Real

IS: now :: thesis: for k being Nat st S_{1}[k] holds

S_{1}[k + 1]

I:
for k being Nat holds SS

let k be Nat; :: thesis: ( S_{1}[k] implies S_{1}[k + 1] )

assume AS: S_{1}[k]
; :: thesis: S_{1}[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 S_{1}[k + 1]
by H0, AS; :: thesis: verum

end;assume AS: S

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 S

ex k being Nat st n = 2 * k by ABIAN:def 2;

hence x |^ n >= 0. F_Real by I; :: thesis: verum