reserve n for Nat,
        p,p1,p2 for Point of TOP-REAL n,
        x for Real;
reserve n,m for non zero Nat;
reserve i,j for Nat;
reserve f for PartFunc of REAL-NS m,REAL-NS n;
reserve g for PartFunc of REAL m,REAL n;
reserve h for PartFunc of REAL m,REAL;
reserve x for Point of REAL-NS m;
reserve y for Element of REAL m;
reserve X for set;

theorem Th5:
for xi be Element of REAL-NS 1 st 1 <= i & i <= j holds
  ||. reproj(i,0.(REAL-NS j)).xi .|| = ||. xi .||
proof
   let xi be Element of REAL-NS 1;
   assume A1: 1 <= i & i <= j;
   consider q being Element of REAL, y being Element of REAL j such that
A2: xi = <*q*> & y = 0.(REAL-NS j)
    & reproj(i,0.(REAL-NS j)).xi = reproj(i,y).q by PDIFF_1:def 6;
A3:reproj(i,0.(REAL-NS j)).xi = Replace(y,i,q) by A2,PDIFF_1:def 5;
    len y = j by CARD_1:def 7;
   then reproj(i,0.(REAL-NS j)).xi = (y| (i-'1))^<*q*>^(y /^ i)
      by A1,A3,FINSEQ_7:def 1;
then A4: ||. reproj(i,0.(REAL-NS j)).xi .|| = |. ( y| (i-'1))^<*q*>^(y/^i) .|
       by A2,REAL_NS1:1;
   y| (i-'1) = (0*j) | (i-'1) by A2,REAL_NS1:def 4;
   then sqrt Sum sqr(y| (i-'1)) = |. 0*(i-'1) .| by A1,Th2;
   then sqrt Sum sqr(y| (i-'1)) = 0 by EUCLID:7;
then A5:Sum sqr(y| (i-'1)) = 0 by RVSUM_1:86,SQUARE_1:24;
    y/^i = (0*j)/^i by A2,REAL_NS1:def 4;
   then sqrt Sum sqr(y/^i) = |. 0*(j-'i) .| by Th3;
then A6:sqrt Sum sqr(y/^i) = 0 by EUCLID:7;
    reconsider q2=q^2 as Real;
    sqr((y| (i-'1))^<*q*>^(y/^i))
     = sqr((y| (i-'1))^<*q*>)^sqr(y/^i) by RVSUM_1:144
    .= sqr(y| (i-'1))^sqr<*q*>^sqr(y/^i) by RVSUM_1:144
    .= sqr(y| (i-'1))^<*q^2*>^sqr(y/^i) by RVSUM_1:55;
   then Sum sqr((y| (i-'1))^<*q*>^(y/^i))
     = Sum(sqr(y| (i-'1))^<*q2*>) + Sum sqr(y/^i) by RVSUM_1:75
    .= Sum sqr(y| (i-'1)) + q^2 + Sum sqr(y/^i) by RVSUM_1:74
    .= q^2 by A5,A6,RVSUM_1:86,SQUARE_1:24;
then A7: ||. reproj(i,0.(REAL-NS j)).xi .|| = |. q .| by A4,COMPLEX1:72;
     proj(1,1).<*q*> = q by PDIFF_1:1;
    hence thesis by A7,A2,PDIFF_1:4;
end;
