
theorem Th29:
  for L being add-associative right_zeroed right_complementable
  Abelian well-unital non degenerated non empty doubleLoopStr for x,z being
  Element of L holds eval(rpoly(1,z),x) = x - z
proof
A1: 2-'1 = 2-1 by XREAL_0:def 2;
A2: 1-'1 = 1-1 by XREAL_0:def 2;
  let L be add-associative right_zeroed right_complementable Abelian
  well-unital non degenerated non empty doubleLoopStr, x,z be Element of L;
  set p = rpoly(1,z);
  consider F be FinSequence of L such that
A3: eval(p,x) = Sum F and
A4: len F = len p and
A5: for n be Element of NAT st n in dom F holds F.n = p.(n-'1) * (power
  L).(x,n-'1) by POLYNOM4:def 2;
A6: deg p = 1 by Th27;
  then
A7: F = <*F.1,F.2*> by A4,FINSEQ_1:44
    .= <*F.1*>^<*F.2*>;
  2 in Seg(len F) by A4,A6;
  then 2 in dom F by FINSEQ_1:def 3;
  then
A8: F.2 = p.1* (power L).(x,1+0) by A5,A1
    .= p.1* ((power L).(x,0) * x) by GROUP_1:def 7
    .= p.1* (1_L * x) by GROUP_1:def 7
    .= p.1* x
    .= 1_L * x by Lm10
    .= x;
  1 in Seg len F by A4,A6,FINSEQ_1:2;
  then 1 in dom F by FINSEQ_1:def 3;
  then F.1 = p.0* (power L).(x,1-'1) by A5,A2
    .= p.0 * 1_L by A2,GROUP_1:def 7
    .= p.0
    .= -power(L).(z,1+0) by Lm10
    .= -(power(L).(z,0) * z) by GROUP_1:def 7
    .= -(1_L * z) by GROUP_1:def 7
    .= -z;
  hence eval(p,x) = Sum(<*-z*>) + Sum(<*x*>) by A3,A7,A8,RLVECT_1:41
    .= Sum(<*-z*>) + x by RLVECT_1:44
    .= -z + x by RLVECT_1:44
    .= x - z by RLVECT_1:def 11;
end;
