let G1, G2, G3, G4 be Ring; :: thesis: for f being Morphism of G1,G2
for g being Morphism of G2,G3
for h being Morphism of G3,G4 st G1 <= G2 & G2 <= G3 & G3 <= G4 holds
h * (g * f) = (h * g) * f
let f be Morphism of G1,G2; :: thesis: for g being Morphism of G2,G3
for h being Morphism of G3,G4 st G1 <= G2 & G2 <= G3 & G3 <= G4 holds
h * (g * f) = (h * g) * f
let g be Morphism of G2,G3; :: thesis: for h being Morphism of G3,G4 st G1 <= G2 & G2 <= G3 & G3 <= G4 holds
h * (g * f) = (h * g) * f
let h be Morphism of G3,G4; :: thesis: ( G1 <= G2 & G2 <= G3 & G3 <= G4 implies h * (g * f) = (h * g) * f )
assume that
A1: G1 <= G2 and
A2: G2 <= G3 and
A3: G3 <= G4 ; :: thesis: h * (g * f) = (h * g) * f
consider f0 being Function of G1,G2 such that
A4: f = RingMorphismStr(# G1,G2,f0 #) by A1, Lm8;
consider h0 being Function of G3,G4 such that
A5: h = RingMorphismStr(# G3,G4,h0 #) by A3, Lm8;
consider g0 being Function of G2,G3 such that
A6: g = RingMorphismStr(# G2,G3,g0 #) by A2, Lm8;
A7: cod g = G3 by A6;
A8: dom h = G3 by A5;
then A9: h * g = RingMorphismStr(# G2,G4,(h0 * g0) #) by A6, A5, A7, Def9;
A10: dom g = G2 by A6;
then A11: dom (h * g) = G2 by A7, A8, Th8;
A12: cod f = G2 by A4;
then A13: cod (g * f) = G3 by A10, A7, Th8;
g * f = RingMorphismStr(# G1,G3,(g0 * f0) #) by A4, A6, A12, A10, Def9;
then h * (g * f) = RingMorphismStr(# G1,G4,(h0 * (g0 * f0)) #) by A5, A8, A13, Def9
.= RingMorphismStr(# G1,G4,((h0 * g0) * f0) #) by RELAT_1:36
.= (h * g) * f by A4, A12, A9, A11, Def9 ;
hence h * (g * f) = (h * g) * f ; :: thesis: verum