
theorem
  for L be add-associative right_zeroed right_complementable unital non
  empty doubleLoopStr for z0,x be Element of L holds eval(<%z0%>,x) = z0
proof
  let L be add-associative right_zeroed right_complementable unital non empty
  doubleLoopStr;
  let z0,x be Element of L;
  consider F be FinSequence of the carrier of L such that
A1: eval(<%z0%>,x) = Sum F and
A2: len F = len <%z0%> and
A3: for n be Element of NAT st n in dom F holds F.n = <%z0%>.(n-'1) * (
  power L).(x,n-'1) by POLYNOM4:def 2;
A4: len F <= 1 by A2,ALGSEQ_1:def 5;
  per cases by A4,NAT_1:25;
  suppose
    len F = 0;
    then
A5: <%z0%> = 0_.(L) by A2,POLYNOM4:5;
    hence eval(<%z0%>,x) = 0.L by POLYNOM4:17
      .= (0_.(L)).0 by FUNCOP_1:7
      .= z0 by A5,Th32;
  end;
  suppose
A6: len F = 1;
    then 0+1 in Seg len F by FINSEQ_1:4;
    then 1 in dom F by FINSEQ_1:def 3;
    then F.1 = <%z0%>.(1-'1) * (power L).(x,1-'1) by A3
      .= <%z0%>.0 * (power L).(x,1-'1) by XREAL_1:232
      .= <%z0%>.0 * (power L).(x,0) by XREAL_1:232
      .= z0 * (power L).(x,0) by Th32
      .= z0 * 1_L by GROUP_1:def 7
      .= z0 by GROUP_1:def 4;
    then F = <*z0*> by A6,FINSEQ_1:40;
    hence thesis by A1,RLVECT_1:44;
  end;
end;
