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 Th19:
  for n being Nat holds
  cdif(f,h).n is Function of V,W
proof
  defpred X[Nat] means cdif(f,h).$1 is Function of V,W;
A1: for k be Nat st X[k] holds X[k+1]
  proof
    let k be Nat;
    assume cdif(f,h).k is Function of V,W;
    then cD(cdif(f,h).k,h) is Function of V,W;
    hence thesis by Def8;
  end;
A2: X[0] by Def8;
  for n be Nat holds X[n] from NAT_1:sch 2(A2,A1);
  hence thesis;
end;
