let D be non empty set ; :: thesis: for i being natural Number
for T being Tuple of i,D
for F being BinOp of D st F is having_a_unity & F is associative & F is having_an_inverseOp holds
( F .: (T,( * T)) = i |-> () & F .: (( * T),T) = i |-> () )

let i be natural Number ; :: thesis: for T being Tuple of i,D
for F being BinOp of D st F is having_a_unity & F is associative & F is having_an_inverseOp holds
( F .: (T,( * T)) = i |-> () & F .: (( * T),T) = i |-> () )

let T be Tuple of i,D; :: thesis: for F being BinOp of D st F is having_a_unity & F is associative & F is having_an_inverseOp holds
( F .: (T,( * T)) = i |-> () & F .: (( * T),T) = i |-> () )

let F be BinOp of D; :: thesis: ( F is having_a_unity & F is associative & F is having_an_inverseOp implies ( F .: (T,( * T)) = i |-> () & F .: (( * T),T) = i |-> () ) )
assume A1: ( F is having_a_unity & F is associative & F is having_an_inverseOp ) ; :: thesis: ( F .: (T,( * T)) = i |-> () & F .: (( * T),T) = i |-> () )
reconsider uT = * T as Element of i -tuples_on D by FINSEQ_2:113;
per cases ( i = 0 or i <> 0 ) ;
suppose A2: i = 0 ; :: thesis: ( F .: (T,( * T)) = i |-> () & F .: (( * T),T) = i |-> () )
then ( F .: (T,uT) = <*> D & F .: (uT,T) = <*> D ) by Lm1;
hence ( F .: (T,( * T)) = i |-> () & F .: (( * T),T) = i |-> () ) by A2; :: thesis: verum
end;
suppose i <> 0 ; :: thesis: ( F .: (T,( * T)) = i |-> () & F .: (( * T),T) = i |-> () )
then reconsider C = Seg i as non empty set ;
T is Function of C,D by Lm4;
hence ( F .: (T,( * T)) = i |-> () & F .: (( * T),T) = i |-> () ) by ; :: thesis: verum
end;
end;