
theorem P5:
for F being Field st Char F <> 2
for a being Element of F
holds ((a + 1.F) / (2 '*' 1.F))^2 - ((a - 1.F) / (2 '*' 1.F))^2 = a
proof
let F be Field; assume AS: Char F <> 2; let a be Element of F;
2 '*' 1.F <> 0.F by AS,P5b;
then A1: (2 '*' 1.F) * (2 '*' 1.F) <>  0.F by VECTSP_1:12;
then C: (2 * 2) '*' 1.F <> 0.F by RING_3:67;
D:  2 '*' 1.F is non zero by AS,P5b;
thus ((a + 1.F) / (2 '*' 1.F))^2 - ((a - 1.F) / (2 '*' 1.F))^2
   = (a + 1.F)^2 / (2 '*' 1.F)^2 - ((a - 1.F) / (2 '*' 1.F))^2 by D,P5a
  .= (a + 1.F)^2 / (2 '*' 1.F)^2 - (a - 1.F)^2 / (2 '*' 1.F)^2 by D,P5a
  .= ((a + 1.F)^2  - (a - 1.F)^2) / (2 '*' 1.F)^2 by A1,VECTSP_2:20
  .= ((a^2 + 2 '*' a * 1.F + (1.F)^2)  - (a - 1.F)^2)
       / (2 '*' 1.F)^2 by P3
  .= ((a^2 + 2 '*' a + (1.F)^2)  - (a^2 - 2 '*' a * 1.F + (1.F)^2))
       / (2 '*' 1.F)^2 by P4
  .= ((a^2 + 2 '*' a + (1.F)^2) + (-(a^2 + -(2 '*' a)) + -(1.F)^2))
       / (2 '*' 1.F)^2 by RLVECT_1:31
  .= ((a^2 + 2 '*' a + (1.F)^2) + (-(a^2) + --(2 '*' a) + -(1.F)^2))
       / (2 '*' 1.F)^2 by RLVECT_1:31
  .= ((a^2 + 2 '*' a) + ((1.F)^2 + ((-(1.F)^2) + (-(a^2) + (2 '*' a)))))
       / (2 '*' 1.F)^2 by RLVECT_1:def 3
  .= ((a^2 + 2 '*' a) + (((1.F)^2 + (-(1.F)^2)) + (-(a^2) + (2 '*' a))))
       / (2 '*' 1.F)^2 by RLVECT_1:def 3
  .= ((a^2 + 2 '*' a) + (0.F + (-(a^2) + (2 '*' a))))
       / (2 '*' 1.F)^2 by RLVECT_1:5
  .= (2 '*' a + ((a^2) + (-(a^2) + (2 '*' a)))) / (2 '*' 1.F)^2
            by RLVECT_1:def 3
  .= (2 '*' a + (((a^2) + -(a^2)) + (2 '*' a))) / (2 '*' 1.F)^2
            by RLVECT_1:def 3
  .= (2 '*' a + (0.F + (2 '*' a))) / (2 '*' 1.F)^2 by RLVECT_1:5
  .= ((2 + 2) '*' a) / (2 '*' 1.F)^2 by RING_3:62
  .= (4 '*' a) / ((2 * 2) '*' 1.F) by RING_3:67
  .= (4 '*' (1.F * a)) / (4 '*' 1.F)
  .= ((4 '*' 1.F) * a) / (4 '*' 1.F) by c1
  .= a / ((4 '*' 1.F) / (4 '*' 1.F)) by C,VECTSP_2:21
  .= a / 1_F by C,VECTSP_2:17
  .= a;
end;
