reserve G for RealNormSpace-Sequence;

theorem Th10:
  for G be RealNormSpace-Sequence, i be Element of dom G,
      x be Point of product G, y be Element of product carr G,
      xi be Point of G.i st y = x
  & xi = y.i holds ||.xi.|| <= ||.x.||
proof
  let G be RealNormSpace-Sequence, i be Element of dom G,
      x be Point of product G, y be Element of product carr G,
      xi be Point of G.i;
  reconsider n = len G as Element of NAT;
  assume that
A1: y=x and
A2: xi=y.i;
A3: (normsequence(G,y).i)^2 = (sqr normsequence(G,y)).i by VALUED_1:11;
A4: for i be Nat st i in Seg n holds 0 <= (sqr normsequence(G,y)) .i
  proof
    let i be Nat such that
    i in Seg n;
    (normsequence(G,y).i)^2 >= 0 by XREAL_1:63;
    hence thesis by VALUED_1:11;
  end;
  dom G = Seg n by FINSEQ_1:def 3; then
A5: 0 <= (normsequence(G,y).i)^2 & (sqr normsequence(G,y)).i <= Sum sqr
  normsequence(G,y) by A4,REAL_NS1:7,XREAL_1:63;
  ||.xi.|| = normsequence(G,y).i by A2,Def11;
  then sqrt ((sqr normsequence(G,y)).i) = normsequence(G,y).i
    by A3,NORMSP_1:4,SQUARE_1:22; then
A6: ||.xi.|| = sqrt ((sqr normsequence(G,y)).i) by A2,Def11;
  ||.x.|| = |.normsequence(G,y).| by A1,Th7;
  hence thesis by A3,A5,A6,SQUARE_1:26;
end;
