let V be Ring_DOMAIN; :: thesis: for g, f being Element of Morphs V st dom g = cod f holds
ex G1, G2, G3 being Element of V st
( G1 <= G2 & G2 <= G3 & g is Morphism of G2,G3 & f is Morphism of G1,G2 )
set X = Morphs V;
defpred S₁[ Element of Morphs V, Element of Morphs V] means dom $1 = cod $2;
let g, f be Element of Morphs V; :: thesis: ( dom g = cod f implies ex G1, G2, G3 being Element of V st
( G1 <= G2 & G2 <= G3 & g is Morphism of G2,G3 & f is Morphism of G1,G2 ) )
assume A1: S₁[g,f] ; :: thesis: ex G1, G2, G3 being Element of V st
( G1 <= G2 & G2 <= G3 & g is Morphism of G2,G3 & f is Morphism of G1,G2 )
consider G2, G3 being Element of V such that
A2: ( G2 <= G3 & g is Morphism of G2,G3 ) by Def17;
consider G1, G29 being Element of V such that
A3: ( G1 <= G29 & f is Morphism of G1,G29 ) by Def17;
A4: G29 = cod f by A3, Def8;
G2 = dom g by A2, Def8;
hence ex G1, G2, G3 being Element of V st
( G1 <= G2 & G2 <= G3 & g is Morphism of G2,G3 & f is Morphism of G1,G2 ) by A1, A2, A3, A4; :: thesis: verum