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
  cdif(r1(#)f1+r2(#)f2,h).(n+1)/.x
  = r1 * cdif(f1,h).(n+1)/.x + r2 * cdif(f2,h).(n+1)/.x
proof
  set g1 = r1(#)f1;
  set g2 = r2(#)f2;
  cdif(r1(#)f1+r2(#)f2,h).(n+1)/.x
  = cdif(g1,h).(n+1)/.x + cdif(g2,h).(n+1)/.x by Th22
  .= r1 * cdif(f1,h).(n+1)/.x + cdif(g2,h).(n+1)/.x by Th21
  .= r1 * cdif(f1,h).(n+1)/.x + r2 * cdif(f2,h).(n+1)/.x by Th21;
  hence thesis;
end;
