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

theorem
  (for Q being multLoop st Q is satisfying_TT satisfying_TL satisfying_TR
     satisfying_LR satisfying_LL satisfying_RR
  holds Q is satisfying_aa1 satisfying_aa2 satisfying_aa3
    satisfying_Ka satisfying_aK1 satisfying_aK2 satisfying_aK3)
  implies
  for Q being AIM multLoop holds
  Q _/_ (lp (Nucl Q)) is commutative multGroup
  &
  Q _/_ (lp (Cent Q)) is multGroup
proof
  assume A1: for Q being multLoop st Q is
     satisfying_TT satisfying_TL satisfying_TR satisfying_LR
     satisfying_LL satisfying_RR
  holds Q is satisfying_aa1 satisfying_aa2 satisfying_aa3
    satisfying_Ka satisfying_aK1 satisfying_aK2 satisfying_aK3;
  let Q be AIM multLoop;
  reconsider Q1 = Q as satisfying_aa1 satisfying_aa2 satisfying_aa3
    satisfying_Ka satisfying_aK1 satisfying_aK2 satisfying_aK3 multLoop by A1;
  set NN = lp (Nucl Q);
  set fN = QuotientHom(Q,NN);
  A2: for y being Element of Q _/_ NN holds ex x being Element of Q st
    fN.x = y
  proof
    let y be Element of Q _/_ NN;
    y in Cosets NN;
    then consider x being Element of Q such that
    A3: y = x * NN by Def41;
    take x;
    thus thesis by A3,Def48;
  end;
    Ker (QuotientHom(Q,NN)) = @ ([#] NN) by Th44;
    then Nucl Q1 c= Ker fN by Th24;
  hence Q _/_ NN is commutative multGroup by Th16,A2;
  set NC = lp (Cent Q);
  set fC = QuotientHom(Q,NC);
  A4: for y being Element of Q _/_ NC holds ex x being Element of Q st
    fC.x = y
  proof
    let y be Element of Q _/_ NC;
    y in Cosets NC;
    then consider x being Element of Q such that
    A5: y = x * NC by Def41;
    fC.x = y by A5,Def48;
    hence thesis;
  end;
  Ker (QuotientHom(Q,NC)) = @ ([#] NC) by Th44;
  then Cent Q1 c= Ker fC by Th25;
  hence Q _/_ NC is multGroup by Th17,A4;
end;
