let F be Field; :: thesis: for a being non zero Element of F
for n, m being Nat holds (anpoly (a,m)) `^ n = anpoly ((a |^ n),(n * m))
let a be non zero Element of F; :: thesis: for n, m being Nat holds (anpoly (a,m)) `^ n = anpoly ((a |^ n),(n * m))
let n, m be Nat; :: thesis: (anpoly (a,m)) `^ n = anpoly ((a |^ n),(n * m))
defpred S<sub>1</sub>[ Nat] means for m being Nat holds (anpoly (a,m)) `^ $1 = anpoly ((a |^ $1),($1 * m));

now :: thesis: for m being Nat holds (anpoly (a,m)) `^ 0 = anpoly ((a |^ 0),(0 * m))

let m be Nat; :: thesis: (anpoly (a,m)) `^ 0 = anpoly ((a |^ 0),(0 * m))
thus (anpoly (a,m)) `^ 0 = 1_. F by POLYNOM5:15
.= (1_ F) | F by RING_4:14
.= anpoly ((1_ F),0) by FIELD_1:7
.= anpoly ((a |^ 0),(0 * m)) by BINOM:8 ; :: thesis: verum

end;

then IA: S₁[ 0 ] ;

IS: now :: thesis: for k being Nat st S₁[k] holds
S₁[k + 1]

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume IV: S₁[k] ; :: thesis: S₁[k + 1]

now :: thesis: for m being Nat holds (anpoly (a,m)) `^ (k + 1) = anpoly ((a |^ (k + 1)),((k + 1) * m))

let m be Nat; :: thesis: (anpoly (a,m)) `^ (k + 1) = anpoly ((a |^ (k + 1)),((k + 1) * m))
thus (anpoly (a,m)) `^ (k + 1) = ((anpoly (a,m)) `^ k) *' (anpoly (a,m)) by POLYNOM5:19
.= (anpoly ((a |^ k),(k * m))) *' (anpoly (a,m)) by IV
.= anpoly (((a |^ k) * a),((k * m) + m)) by FIELD_1:10
.= anpoly (((a |^ k) * (a |^ 1)),((k * m) + m)) by BINOM:8
.= anpoly ((a |^ (k + 1)),((k + 1) * m)) by BINOM:10 ; :: thesis: verum

end;

hence S₁[k + 1] ; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(IA, IS);
hence (anpoly (a,m)) `^ n = anpoly ((a |^ n),(n * m)) ; :: thesis: verum