let m, n be Nat; :: thesis: ( ( m is even or m = 2 * n ) implies (2 |^ m) mod 3 = 1 )
defpred S₁[ Nat] means (2 |^ (2 * $1)) mod 3 = 1;
A1: S₁[ 0 ] by Lm8, NEWTON:4;
A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: S₁[k] ; :: thesis: S₁[k + 1]
A4: 2 * (k + 1) = (2 * k) + 2 ;
((2 |^ (2 * k)) * 4) mod 3 = (1 * 1) mod 3 by A3, Lm10, NAT_D:67
.= 1 by NAT_D:24 ;
hence S₁[k + 1] by A4, Lm2, NEWTON:8; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A1, A2);
hence ( ( m is even or m = 2 * n ) implies (2 |^ m) mod 3 = 1 ) ; :: thesis: verum