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

theorem Th63:
  for Q being AIM multLoop holds
  for x,u being Element of Q holds
    u in Nucl Q implies (x * u) / x in Nucl Q
proof
  let Q be AIM multLoop, x,u be Element of Q;
  assume u in Nucl Q;
  then A1: u in Nucl_l Q & u in Nucl_m Q & u in Nucl_r Q by Th12;
  deffunc Tdx(Element of Q) = (x * $1) / x;
  consider t be Function of Q,Q
  such that
  A2: for v being Element of Q holds t.v = Tdx(v)
  from FUNCT_2:sch 4;
  A3: t in InnAut Q
  proof
    reconsider g = (curry (the multF of Q)).x,
               h = (curry' (the multF of Q)).x as Permutation of Q
      by Th30,Th31;
    A4: t = h" * g
    proof
      for u being Element of Q holds (h * t).u = g.u
      proof
        let u be Element of Q;
        (h * t).u = h.(t.u) by FUNCT_2:15
        .= h.(Tdx(u)) by A2
        .= ((x * u) / x) * x by FUNCT_5:70
        .= g.u by FUNCT_5:69;
        hence thesis;
      end;
      then h"*g = h"*(h*t) by FUNCT_2:def 8
      .= (h"*h)*t by RELAT_1:36
      .= (id the carrier of Q)*t by FUNCT_2:61
      .= t by FUNCT_2:17;
      hence thesis;
    end;
    A5: g in Mlt ([#] Q) by Th32;
    h in Mlt ([#] Q) by Th33;
    then A6: h" in Mlt ([#] Q) by Def35;
    t.(1.Q) = (x * 1.Q) / x by A2
    .= 1.Q by Th6;
    hence thesis by Th55,A6,A4,Def34,A5;
  end;
  for y,z being Element of Q holds (Tdx(u) * y) * z = Tdx(u) * (y * z)
  proof
    let y,z be Element of Q;
    set f = R_MAP(y,z);
    A8: f in InnAut Q by Th58;
    f.u = R_map(u,y,z) by RM1
    .= (u * (y * z)) / (y * z) by Def22,A1
    .= u;
    then Tdx(u) = t.(f.u) by A2
    .= (t*f).u by FUNCT_2:15
    .= (f*t).u by A8,Def50,A3
    .= f.(t.u) by FUNCT_2:15
    .= f.(Tdx(u)) by A2
    .= R_map(Tdx(u),y,z) by RM1
    .= ((Tdx(u) * y) * z) / (y * z);
    hence thesis;
  end;
  then A9: Tdx(u) in Nucl_l Q by Def22;
  for y,z being Element of Q holds (y * z) * Tdx(u) = y * (z * Tdx(u))
  proof
    let y,z be Element of Q;
    set f = L_MAP(z,y);
    f in InnAut Q by Th57;
    then A11: t*f = f*t by Def50,A3;
    f.u = L_map(u,z,y) by LM1
    .= (y * z) \ ((y * z) * u) by Def24,A1
    .= u;
    then Tdx(u) = t.(f.u) by A2
    .= (t*f).u by FUNCT_2:15
    .= f.(t.u) by FUNCT_2:15,A11
    .= f.(Tdx(u)) by A2
    .= L_map(Tdx(u),z,y) by LM1
    .= (y * z) \ (y * (z * Tdx(u)));
    hence thesis;
  end;
  then A12: Tdx(u) in Nucl_r Q by Def24;
  for y,z being Element of Q holds (y * Tdx(u)) * z = y * (Tdx(u) * z)
  proof
    let y,z be Element of Q;
    deffunc M(Element of Q) = y \ ((y * ($1 * z)) / z);
    A13: M(u) = y \ (((y * u) * z) / z) by Def23,A1
    .= u;
    consider m be Function of Q,Q such that
    A14: for v being Element of Q holds m.v = M(v) from FUNCT_2:sch 4;
    A15: m in InnAut Q
    proof
      reconsider h = (curry' (the multF of Q)).(z),
                 k = (curry (the multF of Q)).(y) as Permutation of Q
        by Th31,Th30;
      A16: h in Mlt ([#] Q) & k in Mlt ([#] Q) by Th32,Th33;
      then A17: h" in Mlt ([#] Q) & k" in Mlt ([#] Q) by Def35;
      k*h in Mlt ([#] Q) by A16,Def34;
      then h"*(k*h) in Mlt ([#] Q) by A17,Def34;
      then A18: k"*(h"*(k*h)) in Mlt ([#] Q) by A17,Def34;
      A19: for v being Element of Q holds  (h*k).v = (y * v) * z
      proof
        let v be Element of Q;
        (h*k).v = h.(k.v) by FUNCT_2:15
        .= h.(y * v) by FUNCT_5:69
        .= (y * v) * z by FUNCT_5:70;
        hence thesis;
      end;
      A20: for v being Element of Q holds  (k*h).v = y * (v * z)
      proof
        let v be Element of Q;
        (k*h).v = k.(h.v) by FUNCT_2:15
        .= k.(v * z) by FUNCT_5:70
        .= y * (v * z) by FUNCT_5:69;
        hence thesis;
      end;
      for v being Element of Q holds m.v = (k"*(h"*(k*h))).v
      proof
        let v be Element of Q;
        (y * (m.v)) * z = (y * M(v)) * z by A14
        .= (k*h).v by A20
        .= ((id the carrier of Q)*(k*h)).v by FUNCT_2:17
        .= ((h*h")*(k*h)).v by FUNCT_2:61
        .= (h*(h"*(k*h))).v by RELAT_1:36
        .= (h*((id the carrier of Q)*(h"*(k*h)))).v by FUNCT_2:17
        .= (h*((k*k")*(h"*(k*h)))).v by FUNCT_2:61
        .= (h*(k*(k"*(h"*(k*h))))).v by RELAT_1:36
        .= ((h*k)*(k"*(h"*(k*h)))).v by RELAT_1:36
        .= (h*k).((k"*(h"*(k*h))).v) by FUNCT_2:15
        .= (y * ((k"*(h"*(k*h))).v)) * z by A19;
        then y * (m.v) = y * ((k"*(h"*(k*h))).v) by Th2;
        hence thesis by Th1;
      end;
      then A21: m in Mlt ([#] Q) by A18,FUNCT_2:def 8;
      m.(1.Q) = M(1.Q) by A14
      .= 1.Q by Th5;
      hence thesis by Def49,A21;
    end;
    A22: t*m = m*t by Def50,A15,A3;
    M(Tdx(u)) = m.(Tdx(u)) by A14
    .= m.(t.u) by A2
    .= (t*m).u by A22,FUNCT_2:15
    .= t.(m.u) by FUNCT_2:15
    .= t.(M(u)) by A14
    .= Tdx(u) by A13,A2;
    hence thesis;
  end;
  then Tdx(u) in Nucl_m Q by Def23;
  hence thesis by Th12,A9,A12;
end;
