reserve x, y for set;
reserve D for non empty set;
reserve UN for Universe;
reserve C for Category;
reserve O for non empty Subset of the carrier of C;
reserve G,H for AddGroup;
reserve V for Group_DOMAIN;

theorem Th31:
  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 Def23;
  consider G1,G29 being strict Element of V such that
A3: f is strict Morphism of G1,G29 by Def23;
A4: G29 = cod(f) by A3,Def12;
  G2 = dom(g) by A2,Def12;
  hence thesis by A1,A2,A3,A4;
end;
