
theorem Th56:
  for I being non degenerated domRing-like commutative Ring holds
  canHom(I) is RingHomomorphism
proof
  let I be non degenerated domRing-like commutative Ring;
A1: 0.I <> 1.I;
  for x,y being Element of I holds (canHom(I)).(x+y) = (canHom(I)).x + (
canHom(I)).y & (canHom(I)).(x*y) = (canHom(I)).x * (canHom(I)).y & (canHom(I)).
  (1_I) = 1_the_Field_of_Quotients(I)
  proof
    reconsider res3 = [1.I,1.I] as Element of Q.I by A1,Def1;
    let x,y be Element of I;
    reconsider t1 = (quotient(x,1.I)), t2 = (quotient(y,1.I)) as Element of Q.
    I;
A2: t1`2 = [x,1.I]`2 by A1,Def20
      .= 1.I;
    t1`2 <> 0.I & t2`2 <> 0.I by Th2;
    then
A3: t1`2 * t2`2 <> 0.I by VECTSP_2:def 1;
    then reconsider
    sum = [t1`1*t2`2+t2`1*t1`2,t1`2*t2`2] as Element of Q.I by Def1;
A4: t2`1 = [y,1.I]`1 by A1,Def20
      .= y;
    reconsider prod = [t1`1*t2`1,t1`2*t2`2] as Element of Q.I by A3,Def1;
A5: QClass.t1 = (canHom(I)).x & QClass.t2 = (canHom(I)).y by Def21;
A6: (quotadd(I)).(QClass.t1,QClass.t2) = qadd(QClass.t1,QClass.t2) by Def12
      .= QClass.(padd(t1,t2)) by Th9
      .= QClass.sum;
A7: t1`1 = [x,1.I]`1 by A1,Def20
      .= x;
A8: t2`2 = [y,1.I]`2 by A1,Def20
      .= 1.I;
    then
A9: sum = [t1`1+t2`1*1.I,1.I*1.I] by A2
      .= [t1`1+t2`1,1.I*1.I]
      .= [x+y,1.I] by A4,A7;
    thus (canHom(I)).(x+y) = QClass.(quotient(x+y,1.I)) by Def21
      .= (canHom(I)).x + (canHom(I)).y by A1,A5,A6,A9,Def20;
A10: (quotmult(I)).(QClass.t1,QClass.t2) = qmult(QClass.t1,QClass.t2) by Def13
      .= QClass.(pmult(t1,t2)) by Th10
      .= QClass.prod;
A11: prod = [x*y,1.I] by A8,A2,A4,A7;
    thus (canHom(I)).(x*y) = QClass.(quotient(x*y,1.I)) by Def21
      .= (canHom(I)).x * (canHom(I)).y by A1,A5,A10,A11,Def20;
A12: for u being object holds u in QClass.res3 implies u in q1.I
    proof
      let u be object;
      assume
A13:  u in QClass.res3;
      then reconsider u as Element of Q.I;
      u`1 = u`1 * 1.I
        .= u`1 * res3`2
        .= u`2 * res3`1 by A13,Def4
        .= u`2 * 1.I
        .= u`2;
      hence thesis by Def9;
    end;
    for u being object holds u in q1.I implies u in QClass.res3
    proof
      let u be object;
      assume
A14:  u in q1.I;
      then reconsider u as Element of Q.I;
      u`1 * res3`2 = u`1 * 1.I
        .= u`1
        .= u`2 by A14,Def9
        .= u`2 * 1.I
        .= u`2 * res3`1;
      hence thesis by Def4;
    end;
    then
A15: q1.I = QClass.res3 by A12,TARSKI:2;
    thus (canHom(I)).(1_I) = QClass.(quotient(1.I,1.I)) by Def21
      .= 1_the_Field_of_Quotients(I) by A1,A15,Def20;
  end;
  then canHom(I) is additive multiplicative unity-preserving by GROUP_1:def 13
,GROUP_6:def 6;
  hence thesis;
end;
