let R be preordered Ring; :: thesis: for P being Preordering of R
for n being Nat holds n '*' (1. R) in P
let P be Preordering of R; :: thesis: for n being Nat holds n '*' (1. R) in P
X: P + P c= P by REALALG1:def 14;
defpred S₁[ Nat] means $1 '*' (1. R) in P;
0 '*' (1. R) = 0. R by RING_3:59;
then IA: S₁[ 0 ] by REALALG1:25;

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]
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 S₁[k + 1] by X; :: thesis: verum

end;

I: for k being Nat holds S₁[k] from NAT_1:sch 2(IA, IS);
thus for n being Nat holds n '*' (1. R) in P by I; :: thesis: verum