let D be non empty set ; :: thesis: 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 ; :: thesis: 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; :: thesis: 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; :: thesis: ( 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) ) ; :: thesis: ( T1 = (the_inverseOp_wrt F) * T2 & (the_inverseOp_wrt F) * T1 = T2 )

per cases ( i = 0 or i <> 0 ) ;

suppose i = 0 ; :: thesis: ( T1 = (the_inverseOp_wrt F) * T2 & (the_inverseOp_wrt F) * T1 = T2 )

then ( T1 = <*> D & T2 = <*> D ) ;
hence ( T1 = (the_inverseOp_wrt F) * T2 & (the_inverseOp_wrt F) * T1 = T2 ) ; :: thesis: verum

end;

suppose i <> 0 ; :: thesis: ( T1 = (the_inverseOp_wrt F) * T2 & (the_inverseOp_wrt F) * T1 = T2 )

then reconsider C = Seg i as non empty set ;
( T1 is Function of C,D & T2 is Function of C,D ) by Lm4;
hence ( T1 = (the_inverseOp_wrt F) * T2 & (the_inverseOp_wrt F) * T1 = T2 ) by A1, Th72; :: thesis: verum

end;

end;