let G be Ring; :: thesis: ( 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 ) )
set i = ID G;
thus A1: ( dom (ID G) = G & cod (ID G) = G ) ; :: thesis: ( ( 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 ) )
thus for f being strict RingMorphism st cod f = G holds
(ID G) * f = f :: thesis: for g being strict RingMorphism st dom g = G holds
g * (ID G) = g

proof

let f be strict RingMorphism; :: thesis: ( cod f = G implies (ID G) * f = f )
assume A2: cod f = G ; :: thesis: (ID G) * f = f
set H = dom f;
reconsider f9 = f as Morphism of dom f,G by A2, Th3;
consider m being Function of (dom f),G such that
A3: f9 = RingMorphismStr(# (dom f),G,m #) by A2;
(id G) * m = m by FUNCT_2:17;
hence (ID G) * f = f by A1, A2, A3, Def9; :: thesis: verum

end;

thus for g being strict RingMorphism st dom g = G holds
g * (ID G) = g :: thesis: verum

proof

let f be strict RingMorphism; :: thesis: ( dom f = G implies f * (ID G) = f )
assume A4: dom f = G ; :: thesis: f * (ID G) = f
set H = cod f;
reconsider f9 = f as Morphism of G, cod f by A4, Th3;
consider m being Function of G,(cod f) such that
A5: f9 = RingMorphismStr(# G,(cod f),m #) by A4;
m * (id G) = m by FUNCT_2:17;
hence f * (ID G) = f by A1, A4, A5, Def9; :: thesis: verum

end;