let D be non empty set ; for i being natural Number
for T1, T2 being Tuple of i,D
for F being BinOp of D st F is associative & F is having_an_inverseOp & F is having_a_unity & F .: (T1,T2) = i |-> (the_unity_wrt F) holds
( T1 = (the_inverseOp_wrt F) * T2 & (the_inverseOp_wrt F) * T1 = T2 )
let i be natural Number ; for T1, T2 being Tuple of i,D
for F being BinOp of D st F is associative & F is having_an_inverseOp & F is having_a_unity & F .: (T1,T2) = i |-> (the_unity_wrt F) holds
( T1 = (the_inverseOp_wrt F) * T2 & (the_inverseOp_wrt F) * T1 = T2 )
let T1, T2 be Tuple of i,D; for F being BinOp of D st F is associative & F is having_an_inverseOp & F is having_a_unity & F .: (T1,T2) = i |-> (the_unity_wrt F) holds
( T1 = (the_inverseOp_wrt F) * T2 & (the_inverseOp_wrt F) * T1 = T2 )
let F be BinOp of D; ( F is associative & F is having_an_inverseOp & F is having_a_unity & F .: (T1,T2) = i |-> (the_unity_wrt F) implies ( T1 = (the_inverseOp_wrt F) * T2 & (the_inverseOp_wrt F) * T1 = T2 ) )
assume A1:
( F is associative & F is having_an_inverseOp & F is having_a_unity & F .: (T1,T2) = i |-> (the_unity_wrt F) )
; ( T1 = (the_inverseOp_wrt F) * T2 & (the_inverseOp_wrt F) * T1 = T2 )