reserve L for non empty addLoopStr;
reserve a,b,c,x for Element of L;
reserve L for non empty multLoopStr;
reserve a,b,c,x,y,z for Element of L;
reserve G for multGroup;
reserve a,b,c,x for Element of G;

theorem
  a"*a=1.G & a*(a") = 1.G
proof
  thus
A1: a"*a = 1.G by ALGSTR_0:def 30;
A2: for a,b,c holds (a*b)*c = a*(b*c) by Th11;
  ( for a holds a * 1.G = a)& for a ex x st a*x = 1.G by Th11;
  hence thesis by A1,A2,Lm11;
end;
