
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_1:27
                   .= 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;
