reserve L for non empty addLoopStr;
reserve a,b,c,x for Element of L;

theorem
  (for a holds a + 0.L = a) & (for a ex x st a+x = 0.L) & (for a,b,c
  holds (a+b)+c = a+(b+c)) implies for a ex x st x+a = 0.L
proof
  assume that
A1: for a holds a + 0.L = a and
A2: for a ex x st a+x = 0.L and
A3: for a,b,c holds (a+b)+c = a+(b+c);
  let a;
  consider x such that
A4: a + x = 0.L by A2;
  x+a=0.L by A1,A2,A3,A4,Th1;
  hence thesis;
end;
