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 Th87:
for R being Ring, S being R-monomorphic Ring holds Char S = Char R
proof
let R be Ring, S be R-monomorphic Ring;
set n = Char S, m = Char R;
reconsider n1 = n, m1 = m as Element of INT.Ring by INT_1:def 2;
(the Monomorphism of R,S) * canHom_Int(R) = canHom_Int(S) by Th83;
then ker(canHom_Int(R)) = ker(canHom_Int(S)) by Th69;
then A1: {m1}-Ideal = ker(canHom_Int(S)) by Th81;
then A2: n divides m & m divides n by Th81;
per cases;
suppose A3: m = 0;
  then A4: {0.INT.Ring} = {m1}-Ideal by IDEAL_1:47
                      .= {n1}-Ideal by A1,Th81;
  now assume A5: n <> 0.INT.Ring;
    n1 in {n1}-Ideal by IDEAL_1:66;
    hence contradiction by A5,A4,TARSKI:def 1;
    end;
  hence n = m by A3;
  end;
suppose A6: m > 0;
  consider u being Integer such that A7: m = n * u by A2;
  consider v being Integer such that A8: n = m * v by A2;
  m = (m * v) * u by A7,A8 .= m * (v * u);
  then m/m = (v*u)*(m/m) by XCMPLX_1:74;
  then m/m = (u * v) * 1 by A6,XCMPLX_1:60;
  then A9: u * v = 1 by A6,XCMPLX_1:60;
  u <> -1 by A7,A6;
  then u = 1 & v = 1 by A9,INT_1:9;
  hence thesis by A7;
  end;
end;
