let R be Ring; :: thesis: for E being R -homomorphic Ring
for K being Subring of R
for f being Function of R,E
for g being Function of K,E st g = f | the carrier of K & f is unity-preserving holds
g is unity-preserving
let E be R -homomorphic Ring; :: thesis: for K being Subring of R
for f being Function of R,E
for g being Function of K,E st g = f | the carrier of K & f is unity-preserving holds
g is unity-preserving
let K be Subring of R; :: thesis: for f being Function of R,E
for g being Function of K,E st g = f | the carrier of K & f is unity-preserving holds
g is unity-preserving
let f be Function of R,E; :: thesis: for g being Function of K,E st g = f | the carrier of K & f is unity-preserving holds
g is unity-preserving
let g be Function of K,E; :: thesis: ( g = f | the carrier of K & f is unity-preserving implies g is unity-preserving )
assume that
A1: g = f | the carrier of K and
A2: f is unity-preserving ; :: thesis: g is unity-preserving
thus g . (1_ K) = f . (1_ K) by A1, FUNCT_1:49
.= 1_ E by A2, C0SP1:def 3 ; :: according to GROUP_1:def 13 :: thesis: verum