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

theorem Th60:
  y in x * lp (Nucl Q) iff
  ex u,v st u in Nucl Q & v in Nucl Q & y = u * (x * v)
proof
  thus y in x * lp (Nucl Q) implies ex u,v st u in Nucl Q & v in Nucl Q &
    y = u * (x * v)
  proof
    assume y in x * lp (Nucl Q);
    then y in x * Nucl Q by Th24;
    then consider h being Permutation of the carrier of Q such that
    A1: h in Mlt (Nucl Q) & h.x = y by Def39;
    consider u,v such that
    A2: u in Nucl Q & v in Nucl Q & for z holds h.z = u * (z * v)
      by Th59,A1;
    take u,v;
    thus thesis by A1,A2;
  end;
  given u,v such that
  A3: u in Nucl Q & v in Nucl Q & y = u * (x * v);
  ex h being Permutation of the carrier of Q st h in Mlt (Nucl Q) & h.x = y
  proof
    reconsider h = (curry' (the multF of Q)).(v),
    k = (curry (the multF of Q)).u
    as Permutation of the carrier of Q
    by Th31,Th30;
    take k*h;
    h in Mlt (Nucl Q) & k in Mlt (Nucl Q) by Th33,Th32,A3;
    hence k*h in Mlt (Nucl Q) by Def34;
    (k*h).x = k.(h.x) by FUNCT_2:15
    .= k.(x * v) by FUNCT_5:70
    .= y by A3, FUNCT_5:69;
    hence thesis;
  end;
  then y in x * Nucl Q by Def39;
  hence thesis by Th24;
end;
