
theorem Th8:
for H being RealUnitarySpace,
    G be OrthonormalFamily of H holds
     G is linearly-independent
proof
let H be RealUnitarySpace,
    G be OrthonormalFamily of H;
for l being Linear_Combination of G st Sum l = 0. H holds
Carrier l = {}
proof
let l be Linear_Combination of G;
   assume A1: Sum l = 0. H;
   consider F being FinSequence of H such that
    A2:   F is one-to-one & rng F = Carrier l
           & Sum l = Sum (l (#) F) by RLVECT_2:def 8;
assume A3: Carrier l <> {};
A4:  Carrier l = { v where v is Element of H : l . v <> 0 }
         by RLVECT_2:def 4;
A5:  Carrier l c= G by RLVECT_2:def 6;
consider y be object such that
A6: y in Carrier l by A3,XBOOLE_0:def 1;
consider u be Element of H such that
A7: y=u & l.u <> 0 by A4,A6;
consider n be object such that
A9: n in dom F & u = F.n by FUNCT_1:def 3,A2,A6,A7;
A10: dom F = Seg len F by FINSEQ_1:def 3;
reconsider n as Nat by A9;
A11: u = F/.n by A9,PARTFUN1:def 6;
len (l (#) F) = len F &
    for i being Nat st i in dom (l (#) F) holds
   (l (#) F). i = (l . (F /. i)) * (F /. i)
      by RLVECT_2:def 7; then
A13: 1 <= n & n <= len (l (#) F) by A9,A10,FINSEQ_1:1;
A14: dom (l (#) F) = Seg len (l (#) F) by FINSEQ_1:def 3
    .= Seg len F by RLVECT_2:def 7
    .= dom F by FINSEQ_1:def 3; then
 (l (#) F). n = (l . (F /. n)) * (F /. n)
                  by A9,RLVECT_2:def 7; then
A15: (l (#) F)/. n = (l . (F /. n)) * (F /. n)
   by A14,A9,PARTFUN1:def 6;
  (F/.n) .|. ((l (#) F)/.n)
 = (l . (F /. n)) *((F/.n) .|.(F /. n)) by BHSP_1:3,A15
.= (l . (F /. n)) * 1 by A5,A6,A7,BHSP_5:def 9,A11; then
A17:  (F/.n) .|. ((l (#) F)/.n) <> 0 by A7,A9,PARTFUN1:def 6;
A18:  for i be Nat st 1<=i & i <= len (l (#) F) & n <> i
        holds (F/.n) .|. ((l (#) F)/.i) = 0
proof
  let i be Nat;
  assume A19:1<=i & i <= len (l (#) F) & n <> i; then
  A20a:i in Seg len (l (#) F);
  A20: i in dom (l (#) F) by A19,FINSEQ_3:25; then
 (l (#) F). i = (l . (F /. i)) * (F /. i) by RLVECT_2:def 7; then
  A21: (l (#) F)/. i = (l . (F /. i)) * (F /. i)
   by A20,PARTFUN1:def 6;
A22:i in dom F by A20a,A14,FINSEQ_1:def 3; then
 F.i in rng F by FUNCT_1:def 3; then
F.i in G by A2,A5; then
A23:F/.i in G by A22,PARTFUN1:def 6;
A25: F.i <> F.n by A9,A22,A2,A19;
A27:F/.i <> F/.n by A9,A25,A22,PARTFUN1:def 6,A11;
thus (F/.n) .|. ((l (#) F)/.i)
= (l . (F /. i)) *((F/.n) .|.(F /. i)) by BHSP_1:3,A21
.=(l . (F /. i)) * 0
by A23,A5,A6,A7,A11,A27,BHSP_5:def 8,BHSP_5:def 9
.= 0;
end;
(F/.n) .|. (Sum (l (#) F))
  =(F/.n) .|. ((l (#) F)/.n) by Th5,A13,A18;
hence contradiction by A17,BHSP_1:15,A1,A2;
end;
hence thesis by RLVECT_3:def 1;
end;
