set f = ZeroMap G,H;
let x, y be Element of G; :: according to GRCAT_1:def 13 :: thesis: (ZeroMap G,H) . (x + y) = ((ZeroMap G,H) . x) + ((ZeroMap G,H) . y)
A1: (ZeroMap G,H) . y = 0. H by FUNCOP_1:13;
( (ZeroMap G,H) . (x + y) = 0. H & (ZeroMap G,H) . x = 0. H ) by FUNCOP_1:13;
hence (ZeroMap G,H) . (x + y) = ((ZeroMap G,H) . x) + ((ZeroMap G,H) . y) by A1, RLVECT_1:def 7; :: thesis: verum