let n be Nat; :: thesis: for r being Real holds
( ( r <> 0 or n = 0 ) iff r |^ n <> 0 )
let r be Real; :: thesis: ( ( r <> 0 or n = 0 ) iff r |^ n <> 0 )
defpred S₁[ Nat] means ( ( r <> 0 or $1 = 0 ) iff r |^ $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] )
A2: r |^ (k + 1) = (r |^ k) * r by NEWTON:6;
assume S₁[k] ; :: thesis: S₁[k + 1]
hence S₁[k + 1] by A2; :: thesis: verum

end;

A3: S₁[ 0 ] by NEWTON:4;
for k being Nat holds S₁[k] from NAT_1:sch 2(A3, A1);
hence ( ( r <> 0 or n = 0 ) iff r |^ n <> 0 ) ; :: thesis: verum