let k be Nat; :: thesis: for x being Real st x > 0 holds
x |^ k > 0
let x be Real; :: thesis: ( x > 0 implies x |^ k > 0 )
defpred S₁[ Nat] means for x being Real st x > 0 holds
x |^ $1 > 0 ;
A1: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A2: for x being Real st x > 0 holds
x |^ k > 0 ; :: thesis: S₁[k + 1]
let x be Real; :: thesis: ( x > 0 implies x |^ (k + 1) > 0 )
A3: x |^ (k + 1) = x * (x |^ k) by Th6;
assume A4: x > 0 ; :: thesis: x |^ (k + 1) > 0
then x |^ k > 0 by A2;
hence x |^ (k + 1) > 0 by A4, A3; :: thesis: verum

end;

A5: S₁[ 0 ] by RVSUM_1:94;
for k being Nat holds S₁[k] from NAT_1:sch 2(A5, A1);
hence ( x > 0 implies x |^ k > 0 ) ; :: thesis: verum