reserve p,q for Rational;
reserve g,m,m1,m2,n,n1,n2 for Nat;
reserve i,i1,i2,j,j1,j2 for Integer;
reserve R for Ring, F for Field;

theorem Th104:
for p being Prime, R being p-characteristic Ring,
    i being Integer
holds i '*' 1.R = (i mod p) '*' 1.R
proof
let p be Prime, R be p-characteristic Ring, i be Integer;
Char R = p by Def6; then A1: p '*' 1.R = 0.R by Def5;
A2: ((i div p) * p) '*' 1.R = ((i div p) '*' 1.R) * (p '*' 1.R) by Th66
                          .= 0.R by A1;
thus i '*' 1.R
          = ((i div p) * p + (i mod p)) '*' 1.R by INT_1:59
         .= 0.R + ((i mod p) '*' 1.R) by A2,Th61
         .= (i mod p) '*' 1.R;
end;
