reserve x,y for set;
reserve D for non empty set;
reserve UN for Universe;
reserve f for RingMorphismStr;
reserve G,H,G1,G2,G3,G4 for Ring;
reserve F for RingMorphism;

theorem Th8:
  for f,g being strict RingMorphism st dom g = cod f holds dom(g*f
  ) = dom f & cod (g*f) = cod g
proof
  let f,g be strict RingMorphism;
  assume dom g = cod f;
  then
A1: ex G1,G2,G3 being Ring, f0 being Function of G1,G2, g0 being Function of
G2,G3 st f = RingMorphismStr(#G1,G2,f0#) & g = RingMorphismStr(# G2,G3,g0#) & g
  *f = RingMorphismStr(#G1,G3,g0*f0#) by Th7;
  hence dom(g*f) = dom f;
  thus thesis by A1;
end;
