defpred S₁[ GroupMorphism, GroupMorphism] means dom $1 = cod $2;
let g, f be GroupMorphism; :: thesis: ( dom g = cod f implies ex G1, G2, G3 being AddGroup st
( g is Morphism of G2,G3 & f is Morphism of G1,G2 ) )
assume A1: S₁[g,f] ; :: thesis: ex G1, G2, G3 being AddGroup st
( g is Morphism of G2,G3 & f is Morphism of G1,G2 )
consider G2, G3 being AddGroup such that
A2: g is Morphism of G2,G3 by Th14;
consider G1, G29 being AddGroup such that
A3: f is Morphism of G1,G29 by Th14;
A4: G29 = cod f by A3, Def12;
G2 = dom g by A2, Def12;
hence ex G1, G2, G3 being AddGroup st
( g is Morphism of G2,G3 & f is Morphism of G1,G2 ) by A1, A2, A3, A4; :: thesis: verum