reserve x,y for set;
reserve D for non empty set;
reserve UN for Universe;
reserve R for Ring;
reserve G,H for LeftMod of R;
reserve V for LeftMod_DOMAIN of R;

theorem Th8:
  for g,f being Element of Morphs(V) st dom'(g) = cod'(f) ex G1,G2
,G3 being strict Element of V st g is Morphism of G2,G3 & f is Morphism of G1,
  G2
proof
  set X = Morphs(V);
  defpred P[Element of X,Element of X] means dom'($1) = cod'($2);
  let g,f be Element of X such that
A1: P[g,f];
  consider G2,G3 being strict Element of V such that
A2: g is strict Morphism of G2,G3 by Def7;
  consider G1,G29 being strict Element of V such that
A3: f is strict Morphism of G1,G29 by Def7;
A4: G29 = cod'(f) by A3,MOD_2:def 8;
  G2 = dom'(g) by A2,MOD_2:def 8;
  hence thesis by A1,A2,A3,A4;
end;
