let n be Element of NAT ; :: thesis: for L being non empty non degenerated well-unital domRing-like doubleLoopStr
for x being Element of L st x <> 0. L holds
x |^ n <> 0. L
let L be non empty non degenerated well-unital domRing-like doubleLoopStr ; :: thesis: for x being Element of L st x <> 0. L holds
x |^ n <> 0. L
let x be Element of L; :: thesis: ( x <> 0. L implies x |^ n <> 0. L )
defpred S₁[ Element of NAT ] means x |^ $1 <> 0. L;
assume A1: x <> 0. L ; :: thesis: x |^ n <> 0. L

A2: now

let n be Element of NAT ; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume S₁[n] ; :: thesis: S₁[n + 1]
then (x |^ n) * x <> 0. L by A1, VECTSP_2:def 5;
hence S₁[n + 1] by GROUP_1:def 8; :: thesis: verum

end;

x |^ 0 = 1_ L by BINOM:8;
then A3: S₁[ 0 ] ;
for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A3, A2);
hence x |^ n <> 0. L ; :: thesis: verum