
theorem Th57:
  for I being non degenerated domRing-like commutative Ring holds
  canHom(I) is embedding
proof
  let I be non degenerated domRing-like commutative Ring;
A1: 0.I <> 1.I;
  for x1,x2 being object st x1 in dom canHom(I) & x2 in dom canHom(I) & (
  canHom(I)).x1 = (canHom(I)).x2 holds x1 = x2
  proof
    let x1,x2 be object;
    assume that
A2: x1 in dom canHom(I) & x2 in dom canHom(I) and
A3: (canHom(I)).x1 = (canHom(I)).x2;
    reconsider x1,x2 as Element of I by A2;
    reconsider t1 = quotient(x1,1.I), t2 = quotient(x2,1.I) as Element of Q.I;
A4: t1 in QClass.t1 by Th5;
    QClass.t1 = (canHom(I)).x2 by A3,Def21
      .= QClass.t2 by Def21;
    then
A5: t1`1 * t2`2 = t1`2 * t2`1 by A4,Def4;
A6: t1`2 = [x1,1.I]`2 by A1,Def20
      .= 1.I;
A7: t1`1 = [x1,1.I]`1 by A1,Def20
      .= x1;
A8: t2`1 = [x2,1.I]`1 by A1,Def20
      .= x2;
    t2`2 = [x2,1.I]`2 by A1,Def20
      .= 1.I;
    then x1 = t1`2 * t2`1 by A5,A7
      .= x2 by A6,A8;
    hence thesis;
  end;
  then
A9: (canHom(I)) is one-to-one by FUNCT_1:def 4;
  (canHom(I)) is RingHomomorphism by Th56;
  hence thesis by A9;
end;
