let C, D be non empty set ; :: thesis: for B being Element of Fin C
for d being Element of D
for F, G being BinOp of D
for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F holds
G . d,(F $$ B,f) = F $$ B,(G [;] d,f)
let B be Element of Fin C; :: thesis: for d being Element of D
for F, G being BinOp of D
for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F holds
G . d,(F $$ B,f) = F $$ B,(G [;] d,f)
let d be Element of D; :: thesis: for F, G being BinOp of D
for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F holds
G . d,(F $$ B,f) = F $$ B,(G [;] d,f)
let F, G be BinOp of D; :: thesis: for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F holds
G . d,(F $$ B,f) = F $$ B,(G [;] d,f)
let f be Function of C,D; :: thesis: ( F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F implies G . d,(F $$ B,f) = F $$ B,(G [;] d,f) )
assume that
A1: ( F is commutative & F is associative & F is having_a_unity ) and
A2: F is having_an_inverseOp and
A3: G is_distributive_wrt F ; :: thesis: G . d,(F $$ B,f) = F $$ B,(G [;] d,f)
set e = the_unity_wrt F;
G . d,(the_unity_wrt F) = the_unity_wrt F by A1, A2, A3, FINSEQOP:70;
hence G . d,(F $$ B,f) = F $$ B,(G [;] d,f) by A1, A3, Th14; :: thesis: verum