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
  fdif(r1(#)f1+r2(#)f2,h).(n+1)/.x
  = r1 * (fdif(f1,h).(n+1)/.x) + r2 * (fdif(f2,h).(n+1)/.x)
proof
  set g1 = r1(#)f1;
  set g2 = r2(#)f2;
  reconsider g1n1 = fdif(g1,h).(n+1) as Function of V, W by Th2;
  reconsider g2n1 = fdif(g2,h).(n+1) as Function of V, W by Th2;
  fdif(r1(#)f1+r2(#)f2,h).(n+1)/.x = fdif(g1,h).(n+1)/.x + fdif(g2,h).(n+1)/.x
  by Th8
  .= r1 * (fdif(f1,h).(n+1)/.x) + fdif(g2,h).(n+1)/.x by Th7
  .= r1 * (fdif(f1,h).(n+1)/.x) + r2 * (fdif(f2,h).(n+1)/.x) by Th7;
  hence thesis;
end;
