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 Th9:
  for g,f being Element of Morphs(V) st dom'(g) = cod'(f) holds g*
  f in Morphs(V)
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;
  assume P[g,f];
  then consider G1,G2,G3 being strict Element of V such that
A1: g is Morphism of G2,G3 and
A2: f is Morphism of G1,G2 by Th8;
  reconsider f9 = f as Morphism of G1,G2 by A2;
  reconsider g9 = g as Morphism of G2,G3 by A1;
  g9*f9 = g9*'f9;
  hence thesis by Def7;
end;
