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 Th70:
for R being Ring, S being Subring of R holds R is S-monomorphic
proof
let R be Ring, S be Subring of R;
the carrier of S c= the carrier of R by C0SP1:def 3; then
reconsider f = id S as Function of S,R by FUNCT_2:7;
A1: now let x,y be Element of S;
   A2: [x,y] in [:the carrier of S, the carrier of S:];
   thus f.(x+y) = ((the addF of R)||the carrier of S).(x,y) by C0SP1:def 3
               .= f.x + f.y by A2,FUNCT_1:49;
   end;
A3: now let x,y be Element of S;
   A4: [x,y] in [:the carrier of S, the carrier of S:];
   thus f.(x*y) = ((the multF of R)||the carrier of S).(x,y) by C0SP1:def 3
               .= f.x * f.y by A4,FUNCT_1:49;
   end;
f.(1_S) = 1_R by C0SP1:def 3; then
f is RingHomomorphism by A1,A3,VECTSP_1:def 20,GROUP_1:def 13,GROUP_6:def 6;
hence thesis;
end;
