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
for i be Nat,
    y1,y2 be Point of REAL-NS n holds
  proj(i,n).(y1 + y2) = proj(i,n).y1 + proj(i,n).y2
proof
  let i be Nat,
  y1,y2 be Point of REAL-NS n;
  reconsider yy1 = y1, yy2 = y2 as Element of REAL n by REAL_NS1:def 4;
  reconsider ry1 = yy1.i as Real;
  reconsider ry2 = yy2.i as Real;
A1: proj(i,n).y1 = ry1 & proj(i,n).y2 = ry2 by PDIFF_1:def 1;
    proj(i,n).(y1 + y2) = proj(i,n).(yy1 + yy2) by REAL_NS1:2
                       .= (yy1 + yy2).i by PDIFF_1:def 1
                       .= ry1 + ry2 by RVSUM_1:11;
  hence thesis by A1;
end;
