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 carr f st K,T are_disjoint & a in [#] K & a <> 0. K & not b in [#] K & a1 = a & b1 = b holds
( a * b = the multF of T . ((f . a1),b1) & b * a = the multF of T . (b1,(f . a1)) & not a * b in [#] K & not 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 carr f st K,T are_disjoint & a in [#] K & a <> 0. K & not b in [#] K & a1 = a & b1 = b holds
( a * b = the multF of T . ((f . a1),b1) & b * a = the multF of T . (b1,(f . a1)) & not a * b in [#] K & not 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 carr f st K,T are_disjoint & a in [#] K & a <> 0. K & not b in [#] K & a1 = a & b1 = b holds
( a * b = the multF of T . ((f . a1),b1) & b * a = the multF of T . (b1,(f . a1)) & not a * b in [#] K & not b * a in [#] K )
let a, b be Element of (embField f); :: thesis: for a1, b1 being Element of carr f st K,T are_disjoint & a in [#] K & a <> 0. K & not b in [#] K & a1 = a & b1 = b holds
( a * b = the multF of T . ((f . a1),b1) & b * a = the multF of T . (b1,(f . a1)) & not a * b in [#] K & not b * a in [#] K )
let a1, b1 be Element of carr f; :: thesis: ( K,T are_disjoint & a in [#] K & a <> 0. K & not b in [#] K & a1 = a & b1 = b implies ( a * b = the multF of T . ((f . a1),b1) & b * a = the multF of T . (b1,(f . a1)) & not a * b in [#] K & not b * a in [#] K ) )
assume AS: ( K,T are_disjoint & a in [#] K & a <> 0. K & not b in [#] K & a1 = a & b1 = b ) ; :: thesis: ( a * b = the multF of T . ((f . a1),b1) & b * a = the multF of T . (b1,(f . a1)) & not a * b in [#] K & not b * a in [#] K )
thus C1: a * b = (multemb f) . (a,b) by defemb
.= multemb (f,a1,b1) by defmult, AS
.= the multF of T . ((f . a1),b1) by AS, defmultf ; :: thesis: ( b * a = the multF of T . (b1,(f . a1)) & not a * b in [#] K & not b * a in [#] K )
thus D1: b * a = (multemb f) . (b,a) by defemb
.= multemb (f,b1,a1) by defmult, AS
.= the multF of T . (b1,(f . a1)) by AS, defmultf ; :: thesis: ( not a * b in [#] K & not b * a in [#] K )
the multF of T . ((f . a1),b1) in ([#] T) \ (rng f) by AS, Lm5;
hence not a * b in [#] K by AS, C1, XBOOLE_0:def 4; :: thesis: not b * a in [#] K
the multF of T . (b1,(f . a1)) in ([#] T) \ (rng f) by AS, Lm4;
hence not b * a in [#] K by AS, D1, XBOOLE_0:def 4; :: thesis: verum