reserve L for non empty doubleLoopStr;
reserve a,b,c,x,y,z for Element of L;
reserve G for left-distributive doubleLoop,
  a,b,x,y for Element of G;

theorem
  for G being Abelian left-distributive doubleLoop holds (-1.G)*(-1.G) = 1.G
proof
  let G be Abelian left-distributive doubleLoop;
  thus (-1.G)*(-1.G) = -(1_G*(-1.G)) by Th7
    .= -(-1.G)
    .= 1.G by Th3;
end;
