reserve x for set;
reserve i,j for Integer;
reserve n,n1,n2,n3 for Nat;
reserve p for Prime;
reserve a,b,c,d for Element of GF(p);
reserve K for Ring;
reserve a1,a2,a3,a4,a5,a6 for Element of K;

theorem Th1:
  for F being add-associative right_zeroed right_complementable
      distributive unital non empty doubleLoopStr
  for a being Element of F holds
  a|^2 = (-a)|^2
  proof
   let F be add-associative right_zeroed right_complementable
      distributive unital non empty doubleLoopStr;
   let a be Element of F;
    set a2 = -a;
    thus a|^2 = a*a by GROUP_1:51
    .= a2*a2 by VECTSP_1:10
    .= a2|^2 by GROUP_1:51;
  end;
