reserve i, n for Nat,
  x, y, a for Real,
  v for Element of n-tuples_on REAL,
  p, p1, p2, p3, q, q1, q2 for Point of TOP-REAL n;

theorem
  for y1,y2 being real-valued FinSequence, x1,x2 being Element of REAL
  n st x1=y1 & x2=y2 holds |(y1,y2)|=1/4*((|.(x1+x2).|)^2-(|.(x1-x2).|)^2)
proof
  let y1,y2 be real-valued FinSequence,x1,x2 be Element of REAL n;
  assume
A1: x1=y1 & x2=y2;
  reconsider w1=x1, w2=x2 as Element of n-tuples_on REAL by EUCLID:def 1;
  set v1=sqr (x1+x2), v2=sqr (x1-x2);
  set z1=x1+x2, z2=x1-x2;
A2: 1/4*((|.(x1+x2).|)^2-(|.(x1-x2).|)^2) =1/4*((sqrt Sum sqr (z1))^2-(|.(z2
  ).|)^2) by EUCLID:def 5
    .=1/4*((sqrt Sum sqr (z1))^2-(sqrt Sum sqr (z2))^2) by EUCLID:def 5;
  Sum sqr (x1+x2)>=0 by RVSUM_1:86;
  then
A3: (sqrt Sum sqr (x1+x2))^2 =Sum sqr (x1+x2) by SQUARE_1:def 2;
A4: Sum sqr (x1+x2)-Sum sqr (x1-x2)=Sum (v1 - v2) by RVSUM_1:90;
  Sum sqr (x1-x2)>=0 by RVSUM_1:86;
  then
A5: (sqrt Sum sqr (x1-x2))^2=Sum sqr (x1-x2) by SQUARE_1:def 2;
A6: 2*mlt(w1,w2) +sqr w2+ 2*mlt(w1,w2) = 2*mlt(w1,w2)+(2*mlt(w1,w2) + sqr w2 )
    .= 2*mlt(w1,w2)+2*mlt(w1,w2) + sqr w2 by FINSEQOP:28;
A7: sqr w1 + (2*mlt(w1,w2) + sqr w2) = (2*mlt(w1,w2) + sqr w2) + sqr w1;
  v1-v2=sqr w1 + 2*mlt(w1,w2) + sqr w2 -sqr (w1-w2) by RVSUM_1:68
    .= sqr w1 + 2*mlt(w1,w2) + sqr w2 -(sqr w1 - 2*mlt(w1,w2) + sqr w2) by
RVSUM_1:69
    .= 2*mlt(w1,w2) +sqr w2+ sqr w1 -(sqr w1 - 2*mlt(w1,w2) + sqr w2) by A7,
FINSEQOP:28
    .= 2*mlt(w1,w2) +sqr w2+ sqr w1 -(sqr w1 - 2*mlt(w1,w2))- sqr w2 by
RVSUM_1:39
    .= 2*mlt(w1,w2) +sqr w2+ sqr w1 - sqr w1 + 2*mlt(w1,w2)- sqr w2 by
RVSUM_1:41
    .= (2*mlt(w1,w2) +sqr w2+ 2*mlt(w1,w2))- sqr w2 by RVSUM_1:42
    .= 2*mlt(w1,w2)+2*mlt(w1,w2) by A6,RVSUM_1:42
    .= (2+2)*mlt(w1,w2) by RVSUM_1:50
    .= 4*mlt(w1,w2);
  then 1/4*((|.(x1+x2).|)^2-(|.(x1-x2).|)^2) = 1/4*(4*Sum(mlt(w1,w2))) by A2,A3
,A5,A4,RVSUM_1:87
    .= Sum(mlt(w1,w2));
  hence thesis by A1;
end;
