reserve n for Nat;

theorem Th8:
  for x be Element of REAL n, i be Nat st i in Seg n holds |.x.i.| <= |.x.|
proof
  let x be Element of REAL n;
  let i be Nat;
  reconsider sx = sqr x as Element of REAL n by EUCLID:def 1;
A1: now
    let k be Nat;
    assume k in Seg n;
    sx.k = (x.k)^2 by VALUED_1:11;
    hence 0 <= sx.k by XREAL_1:63;
  end;
A2: 0 <= |.x.i.|*|.x.i.| by XREAL_1:63;
  |.x.i.| * |.x.i.| = (x.i)^2
  proof
    per cases by ABSVALUE:1;
    suppose
      |.x.i.| = x.i;
      hence thesis;
    end;
    suppose
      |.x.i.| = -(x.i);
      hence thesis;
    end;
  end;
  then
A3: (sqr x).i = |.x.i.|*|.x.i.| by VALUED_1:11;
  assume i in Seg n;
  then |.x.i.|*|.x.i.| <= Sum sqr x by A3,A1,Th7;
  then
A4: sqrt (|.x.i.|*|.x.i.|) <= sqrt Sum sqr x by A2,SQUARE_1:26;
  sqrt (|.x.i.|)^2 = |.x.i.| by COMPLEX1:46,SQUARE_1:22;
  hence thesis by A4,EUCLID:def 5;
end;
