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 Th01:
  for f being PartFunc of V, W st x in dom f & x + h in dom f holds
  fD(f,h)/.x = f/.(x+h) - f/.x
proof
  let f be PartFunc of V, W;
  assume
A1: x in dom f & x + h in dom f;
A2: dom Shift(f,h) = -h ++ dom f by Def1;
    (-h) + (x+h) = x + (h+(-h)) by RLVECT_1:def 3
     .= x + 0.V by VECTSP_1:16
     .= x by RLVECT_1:def 4; then
A4: x in (-h) ++ dom f by A1;
A5: Shift(f,h)/.x = Shift(f,h).x by A2,A4,PARTFUN1:def 6
    .= f.(x+h) by Def1,A4
    .= f/.(x+h) by A1,PARTFUN1:def 6;
    x in (dom Shift(f,h)) /\ dom f by A4,A2,A1,XBOOLE_0:def 4;
    then x in dom fD(f,h) by VFUNCT_1:def 2;
    hence fD(f,h)/.x = f/.(x+h) - f/.x by A5,VFUNCT_1:def 2;
end;
