
theorem kercanhomI:
for R being Ring,
    I being Ideal of R holds ker(canHom I) = I
proof
let R be Ring,
    I be Ideal of R;
A: now let xx be object;
   assume AS: xx in ker(canHom I);
   then reconsider x = xx as Element of R;
   Class(EqRel(R,I),0.R) = 0.(R/I) by RING_1:def 6
                        .= (canHom I).x by AS,ker1
                        .= Class(EqRel(R,I),x) by defhomI;
   then x - 0.R in I by RING_1:6;
   hence xx in I;
   end;
now let xx be object;
  assume AS: xx in I;
  then reconsider x = xx as Element of R;
  x - 0.R in I by AS;
  then B: Class(EqRel(R,I),x) = Class(EqRel(R,I),0.R) by RING_1:6;
  (canHom I).x = Class(EqRel(R,I),x) by defhomI
              .= 0.(R/I) by B,RING_1:def 6;
  hence xx in ker(canHom I);
  end;
hence thesis by A,TARSKI:2;
end;
