let D be non empty set ; :: thesis: for F, G being BinOp of D
for i being Nat
for T1, T2 being Element of i -tuples_on D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G = F * (id D),(the_inverseOp_wrt F) holds
G . (F "**" T1),(F "**" T2) = F "**" (G .: T1,T2)
let F, G be BinOp of D; :: thesis: for i being Nat
for T1, T2 being Element of i -tuples_on D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G = F * (id D),(the_inverseOp_wrt F) holds
G . (F "**" T1),(F "**" T2) = F "**" (G .: T1,T2)
let i be Nat; :: thesis: for T1, T2 being Element of i -tuples_on D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G = F * (id D),(the_inverseOp_wrt F) holds
G . (F "**" T1),(F "**" T2) = F "**" (G .: T1,T2)
let T1, T2 be Element of i -tuples_on D; :: thesis: ( F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G = F * (id D),(the_inverseOp_wrt F) implies G . (F "**" T1),(F "**" T2) = F "**" (G .: T1,T2) )
assume that
A1: ( F is commutative & F is associative & F is having_a_unity ) and
A2: ( F is having_an_inverseOp & G = F * (id D),(the_inverseOp_wrt F) ) ; :: thesis: G . (F "**" T1),(F "**" T2) = F "**" (G .: T1,T2)
set e = the_unity_wrt F;
( G . (the_unity_wrt F),(the_unity_wrt F) = the_unity_wrt F & ( for d1, d2, d3, d4 being Element of D holds F . (G . d1,d2),(G . d3,d4) = G . (F . d1,d3),(F . d2,d4) ) ) by A1, A2, FINSEQOP:91, FINSEQOP:94;
hence G . (F "**" T1),(F "**" T2) = F "**" (G .: T1,T2) by A1, Th44; :: thesis: verum