let R be Ring; :: thesis: for G1, G2, G3, G4 being LeftMod of R
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 G1, G2, G3, G4 be LeftMod of R; :: 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 = LModMorphismStr(# G1,G2,f0 #) by Th8;
consider g0 being Function of G2,G3 such that
A2: g = LModMorphismStr(# G2,G3,g0 #) by Th8;
consider h0 being Function of G3,G4 such that
A3: h = LModMorphismStr(# G3,G4,h0 #) by Th8;
A4: h *' g = LModMorphismStr(# G2,G4,(h0 * g0) #) by A2, A3, Th13;
g *' f = LModMorphismStr(# G1,G3,(g0 * f0) #) by A1, A2, Th13;
then h * (g * f) = LModMorphismStr(# G1,G4,(h0 * (g0 * f0)) #) by A3, Th13
.= LModMorphismStr(# G1,G4,((h0 * g0) * f0) #) by RELAT_1:36
.= (h * g) * f by A1, A4, Th13 ;
hence h * (g * f) = (h * g) * f ; :: thesis: verum