reserve n for non zero Element of NAT;
reserve a,b,r,t for Real;
reserve
  Z for open Subset of REAL,
 y0 for VECTOR of REAL-NS n,
  G for Function of REAL-NS n,REAL-NS n;

theorem Th38:
for i be Nat,
   y1 be Point of REAL-NS n, r being Real holds
 proj(i,n).(r*y1) = r*(proj(i,n).y1)
proof
  let i be Nat,
     y1 be Point of REAL-NS n, r be 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).(r*y1) = proj(i,n).(r*yy1) by REAL_NS1:3
    .= (r*yy1).i by PDIFF_1:def 1
    .= r* y1i by RVSUM_1:44;
  hence thesis by PDIFF_1:def 1;
end;
