reserve i,n,m for Nat;

theorem Th26:
for y1,y2 be Point of REAL-NS n holds
 Proj(i,n).(y1 + y2) = Proj(i,n).y1 + Proj(i,n).y2
proof
   let 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 Element of REAL by XREAL_0:def 1;
   reconsider ry2 = yy2.i as Element of REAL by XREAL_0:def 1;
   Proj(i,n).y1 = <* proj(i,n).y1 *> & Proj(i,n).y2 = <* proj(i,n).y2 *>
      by PDIFF_1:def 4; then
A1:Proj(i,n).y1 = <* ry1 *> & Proj(i,n).y2 = <* ry2  *> by PDIFF_1:def 1;
A2:<* ry1 *> is Element of REAL 1 &
   <* ry2 *> is Element of REAL 1 by FINSEQ_2:98;
   Proj(i,n).(y1 + y2) = <* proj(i,n).(y1 + y2) *> by PDIFF_1:def 4
    .= <* proj(i,n).(yy1 + yy2) *> by REAL_NS1:2
    .= <* (yy1 + yy2).i *> by PDIFF_1:def 1
    .= <* (yy1.i + yy2.i) *> by RVSUM_1:11
    .= <* ry1 *> + <* ry2 *> by RVSUM_1:13;
   hence thesis by A1,A2,REAL_NS1:2;
end;
