
theorem spa:
for R being preordered Ring,
    P being Preordering of R for n being Nat holds n '*' 1.R in P
proof
let R be preordered Ring, P be Preordering of R;
X: P + P c= P by REALALG1:def 14;
defpred P[Nat] means $1 '*' 1.R in P;
0 '*' 1.R = 0.R by RING_3:59; then
IA: P[0] by REALALG1:25;
IS: now let k be Nat;
    assume IV: P[k];
    A: (k+1) '*' 1.R = k '*' 1.R + 1 '*' 1.R by RING_3:62
                    .= k '*' 1.R + 1.R by RING_3:60;
    1.R in P by REALALG1:25;
    then (k+1) '*' 1.R in P + P by IV,A;
    hence P[k+1] by X;
    end;
I: for k being Nat holds P[k] from NAT_1:sch 2(IA,IS);
thus thesis by I;
end;
