reserve m,n for non zero Element of NAT;
reserve i,j,k for Element of NAT;
reserve Z for set;

theorem Th11:
for q be Real, i being Nat
 st 1 <= i & i <= j holds |. reproj(i,0*j).q .| = |.q.|
proof
   let q be Real, i be Nat;
   assume A1: 1 <= i & i <= j;
    reconsider q as Element of REAL by XREAL_0:def 1;
   set y = 0*j;
A2:reproj(i,0*j).q = Replace(y,i,q) by PDIFF_1:def 5;
   y in j-tuples_on REAL; then
   ex s be Element of REAL* st s=y & len s = j; then
A3:reproj(i,0*j ).q = (y| (i-'1))^<*q*>^(y/^i) by A1,A2,FINSEQ_7:def 1;
   sqrt Sum sqr(y| (i-'1)) = |. 0*(i-'1) .| by A1,PDIFF_7:2; then
   sqrt Sum sqr(y| (i-'1)) = 0 by EUCLID:7; then
A4:Sum sqr(y| (i-'1)) = 0 by RVSUM_1:86,SQUARE_1:24;
   sqrt Sum sqr(y/^i) = |. 0*(j-'i) .| by PDIFF_7:3; then
A5:sqrt Sum sqr(y/^i) = 0 by EUCLID:7;
   reconsider q2 = q^2 as Element of REAL by XREAL_0:def 1;
   sqr((y| (i-'1))^<*q*>^(y/^i))
     = sqr((y| (i-'1))^<*q*>)^sqr(y/^i) by TOPREAL7:10
    .= sqr(y| (i-'1))^sqr<*q*>^sqr(y/^i) by TOPREAL7:10
    .= 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 A4,A5,RVSUM_1:86,SQUARE_1:24;
   hence thesis by A3,COMPLEX1:72;
end;
