 reserve X,Y,Z,E,F,G,S,T for RealLinearSpace;

theorem NDIFF5def4:
  for G be RealLinearSpace-Sequence,
      i be Element of dom G,
      x be Element of product G holds
  for r being Element of (G.i) holds
  reproj (i,x) . r = x +* (i,r)
  proof
    let G be RealLinearSpace-Sequence,
        i be Element of dom G,
        x be Element of product G;
    ex x0 being Element of product carr G st
    x0 = x & reproj(i,x) = reproj(i,x0) by LOPBAN10:def 2;
    hence thesis by LOPBAN10:def 1;
  end;
