theorem
  for x,y being FinSequence of COMPLEX st len x=len y holds
  |(x-y, x-y)| = |(x, x)| - 2*Re(|(x, y)|) + |(y, y)|
proof
  let x,y be FinSequence of COMPLEX;
  set z=|(x, y)|;
  assume len x=len y;
  then |(x-y, x-y)| =|(x, x)| - |(x, y)| - |(y, x)| + |(y, y)| by Th63
    .=|(x, x)| - |(x, y)| - (|(x, y)|)*' + |(y, y)| by Th64
    .=|(x, x)| - (z + z*') + |(y, y)|
    .=|(x, x)| - 2*Re(|(x, y)|) + |(y, y)| by Th20;
  hence thesis;
end;
