reserve L for non empty doubleLoopStr;
reserve a,b,c,x,y,z for Element of L;

theorem
  for G being Abelian right-distributive doubleLoop, a,x,y being Element
  of G holds a*(x-y) = a*x - a*y
proof
  let G be Abelian right-distributive doubleLoop, a,x,y be Element of G;
  thus a*(x-y) = a*x + a*(-y) by VECTSP_1:def 2
    .= a*x - a*y by Th2;
end;
