
theorem Th44:
  for L be add-associative right_zeroed right_complementable
left-distributive unital non empty doubleLoopStr for z0,z1,x be Element of L
  holds eval(<%z0,z1%>,x) = z0+z1*x
proof
  let L be add-associative right_zeroed right_complementable left-distributive
  unital non empty doubleLoopStr;
  let z0,z1,x be Element of L;
  consider F be FinSequence of the carrier of L such that
A1: eval(<%z0,z1%>,x) = Sum F and
A2: len F = len <%z0,z1%> and
A3: for n be Element of NAT st n in dom F holds F.n = <%z0,z1%>.(n-'1) *
  (power L).(x,n-'1) by POLYNOM4:def 2;
  len F <= 2 by A2,Th39;
  then len F = 0 or ... or len F = 2;
  then per cases;
  suppose
    len F = 0;
    then
A4: <%z0,z1%> = 0_.(L) by A2,POLYNOM4:5;
    hence eval(<%z0,z1%>,x) = 0.L by POLYNOM4:17
      .= (0_.(L)).0 by FUNCOP_1:7
      .= z0 by A4,Th38
      .= z0 + 0.L by RLVECT_1:def 4
      .= z0 + 0.L*x
      .= z0 + (0_.(L)).1*x by FUNCOP_1:7
      .= z0 + z1*x by A4,Th38;
  end;
  suppose
A5: 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,z1%>.(1-'1) * (power L).(x,1-'1) by A3
      .= <%z0,z1%>.0 * (power L).(x,1-'1) by XREAL_1:232
      .= <%z0,z1%>.0 * (power L).(x,0) by XREAL_1:232
      .= z0 * (power L).(x,0) by Th38
      .= z0 * 1_L by GROUP_1:def 7
      .= z0 by GROUP_1:def 4;
    then F = <*z0*> by A5,FINSEQ_1:40;
    hence eval(<%z0,z1%>,x) = z0 by A1,RLVECT_1:44
      .= z0 + 0.L by RLVECT_1:def 4
      .= z0 + 0.L*x
      .= z0 + <%z0,z1%>.1*x by A2,A5,ALGSEQ_1:8
      .= z0 + z1*x by Th38;
  end;
  suppose
A6: len F = 2;
    then 1 in dom F by FINSEQ_3:25;
    then
A7: F.1 = <%z0,z1%>.(1-'1) * (power L).(x,1-'1) by A3
      .= <%z0,z1%>.0 * (power L).(x,1-'1) by XREAL_1:232
      .= <%z0,z1%>.0 * (power L).(x,0) by XREAL_1:232
      .= z0 * (power L).(x,0) by Th38
      .= z0 * 1_L by GROUP_1:def 7
      .= z0 by GROUP_1:def 4;
A8: 2-'1 = 2-1 by XREAL_0:def 2;
    2 in dom F by A6,FINSEQ_3:25;
    then F.2 = <%z0,z1%>.(2-'1) * (power L).(x,2-'1) by A3
      .= z1 * (power L).(x,1) by A8,Th38
      .= z1 * x by GROUP_1:50;
    then F = <*z0,z1*x*> by A6,A7,FINSEQ_1:44;
    hence thesis by A1,RLVECT_1:45;
  end;
end;
