let x be Element of ExtREAL ; :: thesis: for k being Nat st 0 <= x holds
0 <= x |^ k
let k be Nat; :: thesis: ( 0 <= x implies 0 <= x |^ k )
defpred S₁[ Nat] means 0 <= x |^ $1;
assume A1: 0 <= x ; :: thesis: 0 <= x |^ k
A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: S₁[k] ; :: thesis: S₁[k + 1]
x |^ (k + 1) = (x |^ k) * x by Th10;
hence S₁[k + 1] by A1, A3; :: thesis: verum

end;

A4: S₁[ 0 ] by Th6, FINSEQ_2:58;
for k being Nat holds S₁[k] from NAT_1:sch 2(A4, A2);
hence 0 <= x |^ k ; :: thesis: verum