
theorem
  for n be non zero Element of NAT,
      x,y be Element of REAL n,
      i be Element of 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 Element of REAL n, i be Element of NAT;
 thus proj(i,n).(x + y) = (x + y).i by PDIFF_1:def 1
                 .= x.i + y.i by RVSUM_1:11
                 .= proj(i,n).x + y.i by PDIFF_1:def 1
                 .= proj(i,n).x + proj(i,n).y by PDIFF_1:def 1;
end;
