reserve Q,Q1,Q2 for multLoop;
reserve x,y,z,w,u,v for Element of Q;

theorem Th45:
  for Q2 being multLoop holds
  for f being homomorphic Function of Q,Q2 holds
  for h being Function of Q,Q st h in Mlt (Ker f) holds
  f*h = f
proof
  let Q2 be multLoop;
  let f be homomorphic Function of Q,Q2;
  set H = Ker f;
  defpred P[Function of Q,Q] means f * $1 = f;
  A1: for u being Element of Q st u in H holds
  for h being Function of Q,Q st for x being Element of Q holds
    h.x = x * u holds P[h]
  proof
    let u;
    assume A2: u in H;
    let h be Function of Q,Q;
    assume A3: for x holds h.x = x * u;
    P[h]
    proof
      for x holds (f*h).x = f.x
      proof
        let x;
        thus (f*h).x = f.(h.x) by FUNCT_2:15
        .= f.(x * u) by A3
        .= f.x * f.u by Def28b
        .= f.x * 1.Q2 by A2,Def29
        .= f.x;
      end;
      hence thesis by FUNCT_2:def 8;
    end;
    hence thesis;
  end;
  A4: for u being Element of Q st u in H holds
  for h being Function of Q,Q st for x being Element of Q holds
    h.x = u * x holds P[h]
  proof
    let u;
    assume A5: u in H;
    let h be Function of Q,Q;
    assume A6: for x holds h.x = u * x;
    P[h]
    proof
      for x holds (f*h).x = f.x
      proof
        let x;
        thus (f*h).x = f.(h.x) by FUNCT_2:15
        .= f.(u * x) by A6
        .= f.u * f.x by Def28b
        .= 1.Q2 * f.x by A5,Def29
        .= f.x;
      end;
      hence thesis by FUNCT_2:def 8;
    end;
    hence thesis;
  end;
  A7: for g,h being Permutation of Q st P[g] & P[h] holds P[g*h] by RELAT_1:36;
  A8: for g being Permutation of Q st P[g] holds P[g"]
  proof
    let g be Permutation of Q such that A9: P[g];
    P[g"]
    proof
      for x holds (f*(g")).x = f.x
      proof
        let x;
        thus (f*(g")).x = f.((g").x) by FUNCT_2:15
        .= f.(g.((g").x)) by FUNCT_2:15,A9
        .= f.((g*g").x) by FUNCT_2:15
        .= f.((id the carrier of Q).x) by FUNCT_2:61
        .= f.x;
      end;
      hence thesis by FUNCT_2:def 8;
    end;
    hence thesis;
  end;
  for f being Function of Q,Q st f in Mlt H holds P[f]
    from MltInd(A1,A4,A7,A8);
  hence thesis;
end;
