let G1, G2, G3, G4 be AddGroup; :: thesis: for f being strict Morphism of G1,G2
for g being strict Morphism of G2,G3
for h being strict Morphism of G3,G4 holds h * (g * f) = (h * g) * f
let f be strict Morphism of G1,G2; :: thesis: for g being strict Morphism of G2,G3
for h being strict Morphism of G3,G4 holds h * (g * f) = (h * g) * f
let g be strict Morphism of G2,G3; :: thesis: for h being strict Morphism of G3,G4 holds h * (g * f) = (h * g) * f
let h be strict Morphism of G3,G4; :: thesis: h * (g * f) = (h * g) * f
consider f0 being Function of G1,G2 such that
A1: f = GroupMorphismStr(# G1,G2,f0 #) by Th13;
consider g0 being Function of G2,G3 such that
A2: g = GroupMorphismStr(# G2,G3,g0 #) by Th13;
consider h0 being Function of G3,G4 such that
A3: h = GroupMorphismStr(# G3,G4,h0 #) by Th13;
A4: h * g = GroupMorphismStr(# G2,G4,(h0 * g0) #) by A2, A3, Th18;
g * f = GroupMorphismStr(# G1,G3,(g0 * f0) #) by A1, A2, Th18;
then h * (g * f) = GroupMorphismStr(# G1,G4,(h0 * (g0 * f0)) #) by A3, Th18
.= GroupMorphismStr(# G1,G4,((h0 * g0) * f0) #) by RELAT_1:36
.= (h * g) * f by A1, A4, Th18 ;
hence h * (g * f) = (h * g) * f ; :: thesis: verum