
theorem Th11:
  for n be non zero Element of NAT,
      x,y be Point of REAL-NS n,
      i be Nat
  holds Proj(i,n).(x - y) = Proj(i,n).x - Proj(i,n).y
proof
let n be non zero Element of NAT,
    x,y be Point of REAL-NS n,
    i be Nat;
reconsider x1 = x, y1 = y as Element of REAL n by REAL_NS1:def 4;
reconsider rx = x1.i, ry = y1.i as Element of REAL by XREAL_0:def 1;
Proj(i,n).x = <* proj(i,n).x *> & Proj(i,n).y = <* proj(i,n).y *>
  by PDIFF_1:def 4; then
A1: Proj(i,n).x = <* rx *> & Proj(i,n).y = <* ry *> by PDIFF_1:def 1;
A2: <* rx *> is Element of REAL 1 & <* ry *> is Element of REAL 1
      by FINSEQ_2:98;
Proj(i,n).(x - y) = <* proj(i,n).(x - y) *> by PDIFF_1:def 4
  .= <* proj(i,n).(x1 - y1) *> by REAL_NS1:5
  .= <* (x1 - y1).i *> by PDIFF_1:def 1
  .= <* (x1.i - y1.i) *> by RVSUM_1:27
  .= <* rx *> - <* ry *> by RVSUM_1:29;
hence thesis by A1,A2,REAL_NS1:5;
end;
