let R be Ring; :: thesis: for g, f being LModMorphism of R st dom g = cod f holds
ex G1, G2, G3 being LeftMod of R st
( g is Morphism of G2,G3 & f is Morphism of G1,G2 )
defpred S₁[ LModMorphism of R, LModMorphism of R] means dom $1 = cod $2;
let g, f be LModMorphism of R; :: thesis: ( dom g = cod f implies ex G1, G2, G3 being LeftMod of R st
( g is Morphism of G2,G3 & f is Morphism of G1,G2 ) )
assume A1: S₁[g,f] ; :: thesis: ex G1, G2, G3 being LeftMod of R st
( g is Morphism of G2,G3 & f is Morphism of G1,G2 )
consider G2, G3 being LeftMod of R such that
A2: g is Morphism of G2,G3 by Th9;
A3: G2 = dom g by A2, Def8;
consider G1, G2 being LeftMod of R such that
A4: f is Morphism of G1,G2 by Th9;
G2 = cod f by A4, Def8;
hence ex G1, G2, G3 being LeftMod of R st
( g is Morphism of G2,G3 & f is Morphism of G1,G2 ) by A1, A2, A3, A4; :: thesis: verum