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 Th15: :: Triangle
  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;
  then |(x,y)| <= |.|(x,y)|.| & |.|(x,y)|.| <= |.x.|*|.y.| by Th14,ABSVALUE:4;
  then |(x,y)| <= |.x.|*|.y.| by XXREAL_0:2;
  then 2*|(x,y)| <= 2*(|.x.|*|.y.|) by XREAL_1:64;
  then
A2: (|.x.|)^2+ 2*|(x,y)|<=(|.x.|)^2 + 2*(|.x.|*|.y.|) by XREAL_1:7;
  (|.x+y.|)^2 = (|.x.|)^2 + 2*|(y, x)| + (|.y.|)^2 by A1,Th10;
  then
A3: (|.x+y.|)^2 <= (|.x.|)^2 + 2*|.x.|*|.y.|+(|.y.|)^2 by A2,XREAL_1:7;
  0<= (|.x+y.|)^2 by XREAL_1:63;
  then sqrt((|.x+y.|)^2) <= sqrt((|.x.| + |.y.|)^2) by A3,SQUARE_1:26;
  then
A4: |.x+y.| <= sqrt((|.x.| + |.y.|)^2) by EUCLID:9,SQUARE_1:22;
  0<= |.x.| & 0 <= |.y.| by EUCLID:9;
  then 0+0 <= |.x.| + |.y.| by XREAL_1:7;
  hence thesis by A4,SQUARE_1:22;
end;
