
theorem Th2:
  for x,y be FinSequence of REAL st len x = len y &
  for i be Element of NAT st i in Seg len x holds 0 <= x.i & x.i <= y.i
    holds |.x.| <= |.y.|
proof
  let x,y be FinSequence of REAL such that
A1: len x = len y and
A2: for i be Element of NAT st i in Seg len x holds 0 <= x.i & x.i <= y. i;
A3: for i be Nat st i in Seg len x holds (sqr x).i <= (sqr y).i
  proof
    let i be Nat;
    assume i in Seg len x;
    then
A4: 0 <= x.i & x.i <= y.i by A2;
    (x.i)^2=(sqr x).i & (y.i)^2=(sqr y).i by VALUED_1:11;
    hence thesis by A4,SQUARE_1:15;
  end;
  Seg len sqr y = dom sqr y & dom sqr y = dom y by FINSEQ_1:def 3,VALUED_1:11;
  then
A5: len sqr y = len y by FINSEQ_1:def 3;
  Seg len sqr x = dom sqr x & dom sqr x = dom x by FINSEQ_1:def 3,VALUED_1:11;
  then
A6: len sqr x = len x by FINSEQ_1:def 3;
A7: 0 <= Sum sqr x by RVSUM_1:86;
  sqr x is Element of (len sqr x)-tuples_on REAL & sqr y is Element of (
  len sqr y)-tuples_on REAL by FINSEQ_2:92;
  hence thesis by A1,A3,A6,A5,A7,RVSUM_1:82,SQUARE_1:26;
end;
