let V be Ring_DOMAIN; :: thesis: for g, f being Element of Morphs V st dom g = cod f holds
g * f in Morphs V
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 g * f in Morphs V )
assume S₁[g,f] ; :: thesis: g * f in Morphs V
then consider G1, G2, G3 being Element of V such that
A1: ( G1 <= G2 & G2 <= G3 ) and
A2: g is Morphism of G2,G3 and
A3: f is Morphism of G1,G2 by Th19;
reconsider f9 = f as Morphism of G1,G2 by A3;
reconsider g9 = g as Morphism of G2,G3 by A2;
( G1 <= G3 & g9 *' f9 = g9 * f9 ) by A1, Def10, Th5;
hence g * f in Morphs V by Def17; :: thesis: verum