let K be Field; :: thesis: for T being K -monomorphic comRing
for f being Monomorphism of K,T
for a, b being Element of (embField f)
for a1, b1 being Element of T st not a in [#] K & not b in [#] K & the multF of T . (a,b) in rng f & a1 = a & b1 = b holds
( a * b = (f ") . (a1 * b1) & b * a = (f ") . (b1 * a1) & a * b in [#] K & b * a in [#] K )
let T be K -monomorphic comRing; :: thesis: for f being Monomorphism of K,T
for a, b being Element of (embField f)
for a1, b1 being Element of T st not a in [#] K & not b in [#] K & the multF of T . (a,b) in rng f & a1 = a & b1 = b holds
( a * b = (f ") . (a1 * b1) & b * a = (f ") . (b1 * a1) & a * b in [#] K & b * a in [#] K )
let f be Monomorphism of K,T; :: thesis: for a, b being Element of (embField f)
for a1, b1 being Element of T st not a in [#] K & not b in [#] K & the multF of T . (a,b) in rng f & a1 = a & b1 = b holds
( a * b = (f ") . (a1 * b1) & b * a = (f ") . (b1 * a1) & a * b in [#] K & b * a in [#] K )
let a, b be Element of (embField f); :: thesis: for a1, b1 being Element of T st not a in [#] K & not b in [#] K & the multF of T . (a,b) in rng f & a1 = a & b1 = b holds
( a * b = (f ") . (a1 * b1) & b * a = (f ") . (b1 * a1) & a * b in [#] K & b * a in [#] K )
let a1, b1 be Element of T; :: thesis: ( not a in [#] K & not b in [#] K & the multF of T . (a,b) in rng f & a1 = a & b1 = b implies ( a * b = (f ") . (a1 * b1) & b * a = (f ") . (b1 * a1) & a * b in [#] K & b * a in [#] K ) )
assume AS: ( not a in [#] K & not b in [#] K & the multF of T . (a,b) in rng f & a1 = a & b1 = b ) ; :: thesis: ( a * b = (f ") . (a1 * b1) & b * a = (f ") . (b1 * a1) & a * b in [#] K & b * a in [#] K )
reconsider ac = a, bc = b as Element of carr f by defemb;
thus D1: a * b = (multemb f) . (a,b) by defemb
.= multemb (f,ac,bc) by defmult
.= (f ") . (a1 * b1) by AS, defmultf ; :: thesis: ( b * a = (f ") . (b1 * a1) & a * b in [#] K & b * a in [#] K )
C1: the multF of T . (a1,b1) = a1 * b1
.= b1 * a1 by GROUP_1:def 12
.= the multF of T . (b1,a1) ;
thus E1: b * a = (multemb f) . (b,a) by defemb
.= multemb (f,bc,ac) by defmult
.= (f ") . (b1 * a1) by C1, AS, defmultf ; :: thesis: ( a * b in [#] K & b * a in [#] K )
a1 * b1 in dom (f ") by AS, FUNCT_1:33;
then G1: (f ") . (a1 * b1) in rng (f ") by FUNCT_1:def 3;
hence a * b in [#] K by D1; :: thesis: b * a in [#] K
(f ") . (a1 * b1) in dom f by G1, FUNCT_1:33;
then (f ") . (b1 * a1) in dom f by GROUP_1:def 12;
hence b * a in [#] K by E1; :: thesis: verum