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;

theorem Th5:
  cD(f,h)/.x = f/.(x+(2*1.F)"*h) - f/.(x-(2*1.F)"*h)
proof
  dom (Shift(f,(2*1.F)"*h) - Shift(f,-(2*1.F)"*h))
  = the carrier of V by FUNCT_2:def 1;
  hence cD(f,h)/.x = Shift(f,(2*1.F)"*h)/.x - Shift(f,-(2*1.F)"*h)/.x
  by VFUNCT_1:def 2
  .= f/.(x+(2*1.F)"*h) - Shift(f,-(2*1.F)"*h)/.x by Def2
  .= f/.(x+(2*1.F)"*h) - f/.(x-(2*1.F)"*h) by Def2;
end;
