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
  (bdif(f,h).1)/.x = f/.x - Shift(f,-h)/.x
proof
  set f2 = Shift(f,-h);
  (bdif(f,h).1)/.x = bdif(f,h).(0+1)/.x
  .= bD(bdif(f,h).0,h)/.x by Def7
  .= bD(f,h)/.x by Def7
  .= f/.x - f/.(x-h) by Th4
  .= f/.x - f2/.x by Def2;
  hence thesis;
end;
