let D be non empty set ; :: thesis: for F 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 holds

F . ((F "**" T1),(F "**" T2)) = F "**" (F .: (T1,T2))

let F 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 holds

F . ((F "**" T1),(F "**" T2)) = F "**" (F .: (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 holds

F . ((F "**" T1),(F "**" T2)) = F "**" (F .: (T1,T2))

let T1, T2 be Element of i -tuples_on D; :: thesis: ( F is commutative & F is associative & F is having_a_unity implies F . ((F "**" T1),(F "**" T2)) = F "**" (F .: (T1,T2)) )

( len T1 = i & len T2 = i ) by CARD_1:def 7;

hence ( F is commutative & F is associative & F is having_a_unity implies F . ((F "**" T1),(F "**" T2)) = F "**" (F .: (T1,T2)) ) by Th34; :: thesis: verum

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 holds

F . ((F "**" T1),(F "**" T2)) = F "**" (F .: (T1,T2))

let F 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 holds

F . ((F "**" T1),(F "**" T2)) = F "**" (F .: (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 holds

F . ((F "**" T1),(F "**" T2)) = F "**" (F .: (T1,T2))

let T1, T2 be Element of i -tuples_on D; :: thesis: ( F is commutative & F is associative & F is having_a_unity implies F . ((F "**" T1),(F "**" T2)) = F "**" (F .: (T1,T2)) )

( len T1 = i & len T2 = i ) by CARD_1:def 7;

hence ( F is commutative & F is associative & F is having_a_unity implies F . ((F "**" T1),(F "**" T2)) = F "**" (F .: (T1,T2)) ) by Th34; :: thesis: verum