 reserve a for non empty set;
 reserve b, x, o for object;
reserve R for right_zeroed add-associative right_complementable Abelian
  well-unital distributive associative non trivial non trivial doubleLoopStr;

theorem
    for m be Element of R holds eval(1_1(R),m) = m
    proof
      let m be Element of R;
A1:   len 1_1(R) = 1+1 by NIVEN:20;
A2:   1 in dom(0_.(R));
      eval(1_1(R),m) = eval (Leading-Monomial(1_1(R)),m) by Th20
      .= (1_1(R)).(len(1_1(R))-'1)*(power R).(m,len(1_1(R))-'1) by POLYNOM4:22
      .= 1.R * (power R).(m,1) by A1,A2,FUNCT_7:31
      .= (power R).(m,0+1)
      .= power(R).(m,0) * m by GROUP_1:def 7
      .= 1_R * m by GROUP_1:def 7
      .= m;
      hence thesis;
    end;
