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 Th14: :: Schwartz
  for x,y being real-valued FinSequence st len x=len y holds
  |.|(x,y)|.| <= |.x.|*|.y.|
proof
  let x,y be real-valued FinSequence;
  assume
A1: len x=len y;
A2: x = 0*(len x) implies |.|(x,y)|.| <= sqrt (|(x,x)|) * sqrt (|(y,y)|)
  proof
    assume x = 0*(len x);
    then |(x,y)| = 0 & |(x,x)|=0 by A1,Th8;
    hence thesis by ABSVALUE:2;
  end;
A3: x <> 0*(len x) implies |.|(x,y)|.| <= sqrt (|(x,x)|) * sqrt (|(y,y)|)
  proof
A4: for t be Real holds |(x,x)| * t^2 + (2 * (|(x,y)|)) * t + |(y,
    y)| >= 0
    proof
      let t be Real;
      set s=t^2;
A5:   len (t * x)=len x by RVSUM_1:117;
      |((t * x + y),(t * x + y))| >= 0 by RVSUM_1:119;
      then
      |((t * x) , (t * x))| + 2 * (|((t*x),y)|) + |(y,y)| >= 0 by A1,A5,
RVSUM_1:128;
      then t * ( |(t*x,x)|) + 2 * |((t*x),y)| + |(y,y)| >= 0 by A5,RVSUM_1:121;
      then t * ( t * |(x,x)|) + 2 * |((t*x),y)| + |(y,y)| >= 0 by A1,
RVSUM_1:121;
      then |(x,x)| * s + 2 * (|(x,y)| * t) + |(y,y)| >= 0 by A1,RVSUM_1:121;
      hence thesis;
    end;
    set w=|.|(x,y)|.|, u=|(x,y)|;
A6: |(x,x)| >= 0 by RVSUM_1:119;
    assume x <> 0*(len x);
    then |(x,x)| <> 0 by Th6;
    then |(x,x)| > 0 by A6,XXREAL_0:1;
    then delta(|(x,x)|,(2 * (|(x,y)|)),|(y,y)|) <= 0 by A4,QUIN_1:10;
    then (2 * u)^2 - 4 * (|(x,x)|) * (|(y,y)|) <= 0 by QUIN_1:def 1;
    then 4 * ((( u )^2) - (|(x,x)|) * (|(y,y)|)) <= 0;
    then ((|(x,y)|)^2) - (|(x,x)|) * (|(y,y)|) <= 0/4 by XREAL_1:77;
    then (|(x,y)|)^2 <= (|(x,x)|) * (|(y,y)|) by XREAL_1:50;
    then 0 <= w^2 & (|.|(x,y)|.|)^2 <= (|(x,x)|) * (|(y,y)|) by COMPLEX1:75
,XREAL_1:63;
    then (sqrt (|.|(x,y)|.|)^2) <= sqrt ((|(x,x)|) * (|(y,y)|)) by SQUARE_1:26;
    then
A7: |.|(x,y)|.| <= sqrt (|(x,x)| * |(y,y)|) by COMPLEX1:46,SQUARE_1:22;
    |(y,y)| >= 0 by RVSUM_1:119;
    hence thesis by A6,A7,SQUARE_1:29;
  end;
  sqrt (|(x,x)|)=|.x.| by Th5;
  hence thesis by A2,A3,Th5;
end;
