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 S_{1}[ Nat] means ( ( r <> 0 or $1 = 0 ) iff r |^ $1 <> 0 );

A1: for k being Nat st S_{1}[k] holds

S_{1}[k + 1]
_{1}[ 0 ]
by NEWTON:4;

for k being Nat holds S_{1}[k]
from NAT_1:sch 2(A3, A1);

hence ( ( r <> 0 or n = 0 ) iff r |^ n <> 0 ) ; :: thesis: verum

( ( r <> 0 or n = 0 ) iff r |^ n <> 0 )

let r be Real; :: thesis: ( ( r <> 0 or n = 0 ) iff r |^ n <> 0 )

defpred S

A1: for k being Nat st S

S

proof

A3:
S
let k be Nat; :: thesis: ( S_{1}[k] implies S_{1}[k + 1] )

A2: r |^ (k + 1) = (r |^ k) * r by NEWTON:6;

assume S_{1}[k]
; :: thesis: S_{1}[k + 1]

hence S_{1}[k + 1]
by A2; :: thesis: verum

end;A2: r |^ (k + 1) = (r |^ k) * r by NEWTON:6;

assume S

hence S

for k being Nat holds S

hence ( ( r <> 0 or n = 0 ) iff r |^ n <> 0 ) ; :: thesis: verum