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
  dom ID(G) = G & cod ID(G) = G & (for f being strict RingMorphism
st cod f = G holds (ID G)*f = f) & for g being strict RingMorphism st dom g = G
  holds g*(ID G) = g
proof
  set i = ID G;
  thus
A1: dom i = G & cod i = G;
  thus for f being strict RingMorphism st cod f = G holds i*f = f
  proof
    let f be strict RingMorphism such that
A2: cod f = G;
    set H = dom(f);
    reconsider f9 = f as Morphism of H,G by A2,Th3;
    consider m being Function of H,G such that
A3: f9 = RingMorphismStr(#H,G,m#) by A2;
    (id G)*m = m by FUNCT_2:17;
    hence thesis by A1,A2,A3,Def9;
  end;
  thus for g being strict RingMorphism st dom g = G holds g*(ID G) = g
  proof
    let f be strict RingMorphism such that
A4: dom f = G;
    set H = cod(f);
    reconsider f9 = f as Morphism of G,H by A4,Th3;
    consider m being Function of G,H such that
A5: f9 = RingMorphismStr(#G,H,m#) by A4;
    m*(id G) = m by FUNCT_2:17;
    hence thesis by A1,A4,A5,Def9;
  end;
end;
