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 Th4:
  bD(f,h)/.x = f/.x - f/.(x-h)
proof
P1:dom f = the carrier of V by FUNCT_2:def 1;
  dom Shift(f,-h) = the carrier of V by FUNCT_2:def 1;
  then x in (dom f) /\ (dom Shift(f,-h)) by P1; then
P2: x in dom (f - Shift(f,-h)) by VFUNCT_1:def 2;
  thus bD(f,h)/.x = f/.x - (Shift(f,-h))/.x by P2, VFUNCT_1:def 2
  .= f/.x - f/.(x-h) by Def2;
end;
