
theorem Th29:
  for V being RealUnitarySpace, x,y being VECTOR of V holds ||.x +
y.||^2 = ||.x.||^2 + 2 * x .|. y + ||.y.||^2 & ||.x - y.||^2 = ||.x.||^2 - 2 *
  x .|. y + ||.y.||^2
proof
  let V be RealUnitarySpace;
  let x,y be VECTOR of V;
A1: x .|. x >= 0 by BHSP_1:def 2;
  ||.x + y.|| = sqrt ((x + y) .|. (x + y)) by BHSP_1:def 4;
  then
A2: sqrt ||.x + y.||^2 = sqrt ((x + y) .|. (x + y)) by BHSP_1:28,SQUARE_1:22;
A3: y .|. y >= 0 by BHSP_1:def 2;
  (x + y) .|. (x + y) >= 0 & ||.x + y.||^2 >= 0 by BHSP_1:def 2,XREAL_1:63;
  then ||.x + y.||^2 = (x + y) .|. (x + y) by A2,SQUARE_1:28
    .= x .|. x + 2 * x .|. y + y .|. y by BHSP_1:16
    .= (sqrt (x .|. x))^2 + 2 * x .|. y + y .|. y by A1,SQUARE_1:def 2
    .= ||.x.||^2 + 2 * x .|. y + y .|. y by BHSP_1:def 4
    .= ||.x.||^2 + 2 * x .|. y + (sqrt (y .|. y))^2 by A3,SQUARE_1:def 2;
  hence ||.x + y.||^2 = ||.x.||^2 + 2 * x .|. y + ||.y.||^2 by BHSP_1:def 4;
  ||.x - y.|| = sqrt ((x - y) .|. (x - y)) by BHSP_1:def 4;
  then
A4: sqrt ||.x - y.||^2 = sqrt ((x - y) .|. (x - y)) by BHSP_1:28,SQUARE_1:22;
  (x - y) .|. (x - y) >= 0 & ||.x - y.||^2 >= 0 by BHSP_1:def 2,XREAL_1:63;
  then ||.x - y.||^2 = (x - y) .|. (x - y) by A4,SQUARE_1:28
    .= x .|. x - 2 * x .|. y + y .|. y by BHSP_1:18
    .= (sqrt (x .|. x))^2 - 2 * x .|. y + y .|. y by A1,SQUARE_1:def 2
    .= ||.x.||^2 - 2 * x .|. y + y .|. y by BHSP_1:def 4
    .= ||.x.||^2 - 2 * x .|. y + (sqrt (y .|. y))^2 by A3,SQUARE_1:def 2;
  hence thesis by BHSP_1:def 4;
end;
