reserve i,n,m for Nat;

theorem Th27:
for y1 be Point of REAL-NS n, r being Real holds
 Proj(i,n).(r*y1) = r*(Proj(i,n).y1)
proof
   let y1 be Point of REAL-NS n, r being Real;
   reconsider yy1 = y1 as Element of REAL n by REAL_NS1:def 4;
   reconsider y1i = yy1.i as Element of REAL by XREAL_0:def 1;
   Proj(i,n).y1 = <* proj(i,n).y1 *> by PDIFF_1:def 4; then
A1:Proj(i,n).y1 = <* y1i *> by PDIFF_1:def 1;
A2:<* y1i *> is Element of REAL 1 by FINSEQ_2:98;
   Proj(i,n).(r*y1) = <* proj(i,n).(r*y1) *> by PDIFF_1:def 4
     .= <* proj(i,n).(r*yy1) *> by REAL_NS1:3
     .= <* (r*yy1).i *> by PDIFF_1:def 1
     .= <* r*(yy1.i) *> by RVSUM_1:44
     .= r*<* y1i *> by RVSUM_1:47;
   hence thesis by A1,A2,REAL_NS1:3;
end;
