{} V = {} ;
then reconsider P = {} as finite Subset of V ;
reconsider f = the carrier of V --> 0c as Function of the carrier of V,COMPLEX ;
reconsider f = f as Element of Funcs the carrier of V,COMPLEX by FUNCT_2:11;
take f ; :: thesis: ex T being finite Subset of V st
for v being Element of V st not v in T holds
f . v = 0
take P ; :: thesis: for v being Element of V st not v in P holds
f . v = 0
thus for v being Element of V st not v in P holds
f . v = 0 by FUNCOP_1:13; :: thesis: verum