reserve C for non empty set;
reserve GF for Field,
        V for VectSp of GF,
        v,u for Element of V,
        W for Subset of V;
reserve f,f1,f2,f3 for PartFunc of C,V;
reserve F,G for Field,
        V for VectSp of F,
        W for VectSp of G;
reserve f,f1,f2 for Function of V, W;
reserve x,h for Element of V;
reserve r,r1,r2 for Element of G;
reserve n,m,k for Nat;

theorem
  (cdif(f,h).1)/.x = Shift(f,((2*1.F)"*h))/.x - Shift(f,-((2*1.F)"*h))/.x
proof
  set f2 = Shift(f,-((2*1.F)"*h));
  set f1 = Shift(f,((2*1.F)"*h));
  (cdif(f,h).1)/.x = cdif(f,h).(0+1)/.x
  .= cD(cdif(f,h).0,h)/.x by Def8
  .= cD(f,h)/.x by Def8
  .= f/.(x+(2*1.F)"*h) - f/.(x-(2*1.F)"*h) by Th5
  .= f1/.x - f/.(x+-((2*1.F)"*h)) by Def2
  .= f1/.x - f2/.x by Def2;
  hence thesis;
end;
