 reserve j for set;
 reserve p,r for Real;
 reserve S,T,F for RealNormSpace;
 reserve x0 for Point of S;
 reserve g for PartFunc of S,T;
 reserve c for constant sequence of S;
 reserve R for RestFunc of S,T;
 reserve G for RealNormSpace-Sequence;
 reserve i for Element of dom G;
 reserve f for PartFunc of product G,F;
 reserve x for Element of product G;
reserve G for RealNormSpace-Sequence;
reserve F for RealNormSpace;
reserve i for Element of dom G;
reserve f,f1,f2 for PartFunc of product G, F;
reserve x for Point of product G;
reserve X for set;

theorem Th21:
for xi be Element of G.i
  holds ||. reproj(i,0.(product G)).xi .|| = ||.xi.||
proof
   let xi be Element of G.i;
   set j = len G;
   reconsider i0 = i as Element of NAT;
   Seg len G = dom G by FINSEQ_1:def 3; then
A1:1 <= i0 & i0 <= j by FINSEQ_1:1;
   set z = 0.(product G);
A3: the carrier of (product G) = product carr G by Th10;
   then reconsider w = z +* (i0,xi) as Element of product carr G by Th11;
A4: ||. reproj(i,z).xi .|| = |. normsequence(G,w) .|
         by Def4,PRVECT_2:7;
   reconsider q = ||.xi.|| as Element of REAL;
   set q1 = <*q*>;
   set y = 0*j;
A5:len normsequence(G,w) = j by PRVECT_2:def 11;
A6:len y = j by CARD_1:def 7; then
A7:(y| (i0-'1))^<*q*>^(y /^ i0) = Replace(y,i0,q) by A1,FINSEQ_7:def 1; then
A8:len ((y| (i0-'1))^<*q*>^(y /^ i0)) = len y by FINSEQ_7:5;
A9:len y = len Replace(y,i0,q) by FINSEQ_7:5;
   for k be Nat st 1 <= k & k <= len normsequence(G,w) holds
    normsequence(G,w).k = ((y| (i0-'1))^<*q*>^(y /^ i0)).k
   proof
    let k be Nat;
    assume A10: 1 <= k & k <= len normsequence(G,w); then
    reconsider k1 = k as Element of dom G by A5,FINSEQ_3:25;
A11: k1 in dom G;
    z in the carrier of product G; then
    z in product carr G by Th10; then
    consider g being Function such that
A12: z = g & dom g = dom carr G &
      for y being object st y in dom carr G holds g.y in (carr G).y
        by CARD_3:def 5;
A13: k in dom z by A11,A12,Lm1;
A14: (normsequence(G,w)).k
      = (the normF of (G.k1)).(w.k1) by PRVECT_2:def 11;
    per cases;
    suppose A15: k = i0; then
A16:  (normsequence(G,w)).k = ||. xi .|| by A14,A13,FUNCT_7:31;
     Replace(y,i0,q)/.k = q by A15,A10,A5,A6,FINSEQ_7:8;
     hence normsequence(G,w).k = ((y| (i0-'1))^<*q*>^(y /^ i0)).k
        by A16,A7,A10,A5,A6,A9,FINSEQ_4:15;
    end;
    suppose A17: k <> i0; then
     w.k1 = z.k1 by FUNCT_7:32; then
A18:  (normsequence(G,w)).k = ||. 0.(G.k1) .|| by A14,Th14,A3;
     Replace(y,i0,q)/.k = y/.k by A17,A10,A5,A6,FINSEQ_7:10; then
     Replace(y,i0,q).k = y/.k by A10,A5,A6,A9,FINSEQ_4:15; then
     Replace(y,i0,q).k = y.k by A10,A5,A6,FINSEQ_4:15;
     hence normsequence(G,w).k = ((y| (i0-'1))^<*q*>^(y /^ i0)).k
       by A18,A6,A1,FINSEQ_7:def 1;
    end;
   end; then
A19:normsequence(G,w) = (y| (i0-'1))^<*q*>^(y /^ i0)
     by A6,A8,PRVECT_2:def 11;
   sqrt Sum sqr(y| (i0-'1)) = |. 0*(i0-'1) .| by A1,PDIFF_7:2; then
   sqrt Sum sqr(y| (i0-'1)) = 0 by EUCLID:7; then
A20:Sum sqr(y| (i0-'1)) = 0 by RVSUM_1:86,SQUARE_1:24;
   sqrt Sum sqr(y/^i0) = |. 0*(j-'i0) .| by PDIFF_7:3; then
A21:sqrt Sum sqr(y/^i0) = 0 by EUCLID:7;
   reconsider q2 = q^2 as Element of REAL by XREAL_0:def 1;
   sqr((y| (i0-'1))^<*q*>^(y/^i0))
     = sqr((y| (i0-'1))^<*q*>)^sqr(y/^i0) by RVSUM_1:144
    .= sqr(y| (i0-'1))^sqr<*q*>^sqr(y/^i0) by RVSUM_1:144
    .= sqr(y| (i0-'1))^<*q^2*>^sqr(y/^i0) by RVSUM_1:55; then
   Sum sqr((y| (i0-'1))^<*q*>^(y/^i0))
     = Sum(sqr(y| (i0-'1))^<*q2*>) + Sum sqr(y/^i0) by RVSUM_1:75
    .= Sum sqr(y| (i0-'1)) + q^2 + Sum sqr(y/^i0) by RVSUM_1:74
    .= q^2 by A20,A21,RVSUM_1:86,SQUARE_1:24; then
   ||. reproj(i,z).xi .|| = |. q .| by A19,A4,COMPLEX1:72;
   hence thesis by COMPLEX1:43;
end;
