theorem Th15: :: ANPROJ_2:15

for A being non empty set
for x1, x2, x3 being Element of A st A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 holds
ex f, g, h being Element of Funcs (A,REAL) st
( ( for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 ) ) & ( for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) ) )