let UN be Universe; :: thesis: for R being Ring
for f, g, h being Morphism of st dom h = cod g & dom g = cod f holds
h * (g * f) = (h * g) * f
let R be Ring; :: thesis: for f, g, h being Morphism of st dom h = cod g & dom g = cod f holds
h * (g * f) = (h * g) * f
set X = Morphs (LModObjects UN,R);
let f, g, h be Morphism of ; :: thesis: ( dom h = cod g & dom g = cod f implies h * (g * f) = (h * g) * f )
assume A1: ( dom h = cod g & dom g = cod f ) ; :: thesis: h * (g * f) = (h * g) * f
reconsider f' = f, g' = g, h' = h as strict Element of Morphs (LModObjects UN,R) by Th15;
A2: ( h' * g' = h * g & dom (h * g) = cod f ) by A1, Lm1, Th18;
A3: ( dom h' = cod g' & dom g' = cod f' ) by A1, Th18;
then reconsider gf = g' * f', hg = h' * g' as strict Element of Morphs (LModObjects UN,R) by Th12;
( g' * f' = g * f & dom h = cod (g * f) ) by A1, Lm1, Th18;
then h * (g * f) = h' * gf by Th18
.= hg * f' by A3, MOD_2:25
.= (h * g) * f by A2, Th18 ;
hence h * (g * f) = (h * g) * f ; :: thesis: verum