
theorem Th29:
  for L be add-associative right_zeroed right_complementable
well-unital right-distributive non empty doubleLoopStr for v be Element of L
  holds v*1_.(L) = <%v%>
proof
  let L be add-associative right_zeroed right_complementable well-unital
  right-distributive non empty doubleLoopStr;
  let v be Element of L;
  now
    let n be Element of NAT;
    per cases;
    suppose
A1:   n=0;
      hence <%v%>.n = v by ALGSEQ_1:def 5
        .= v*1.L
        .= v*(1_.(L)).n by A1,POLYNOM3:30;
    end;
    suppose
A2:   n<>0;
A3:   len <%v%> <= 1 by ALGSEQ_1:def 5;
      n >= 0+1 by A2,NAT_1:13;
      hence <%v%>.n = 0.L by A3,ALGSEQ_1:8,XXREAL_0:2
        .= v*0.L
        .= v*(1_.(L)).n by A2,POLYNOM3:30;
    end;
  end;
  hence thesis by Def4;
end;
