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,((the_inverseOp_wrt F) * T)) = i |-> (the_unity_wrt F) & F .: (((the_inverseOp_wrt F) * T),T) = i |-> (the_unity_wrt F) )
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,((the_inverseOp_wrt F) * T)) = i |-> (the_unity_wrt F) & F .: (((the_inverseOp_wrt F) * T),T) = i |-> (the_unity_wrt F) )
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,((the_inverseOp_wrt F) * T)) = i |-> (the_unity_wrt F) & F .: (((the_inverseOp_wrt F) * T),T) = i |-> (the_unity_wrt F) )
let F be BinOp of D; :: thesis: ( F is having_a_unity & F is associative & F is having_an_inverseOp implies ( F .: (T,((the_inverseOp_wrt F) * T)) = i |-> (the_unity_wrt F) & F .: (((the_inverseOp_wrt F) * T),T) = i |-> (the_unity_wrt F) ) )
assume A1: ( F is having_a_unity & F is associative & F is having_an_inverseOp ) ; :: thesis: ( F .: (T,((the_inverseOp_wrt F) * T)) = i |-> (the_unity_wrt F) & F .: (((the_inverseOp_wrt F) * T),T) = i |-> (the_unity_wrt F) )
reconsider uT = (the_inverseOp_wrt F) * 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,((the_inverseOp_wrt F) * T)) = i |-> (the_unity_wrt F) & F .: (((the_inverseOp_wrt F) * T),T) = i |-> (the_unity_wrt F) )

then ( F .: (T,uT) = <*> D & F .: (uT,T) = <*> D ) by Lm1;
hence ( F .: (T,((the_inverseOp_wrt F) * T)) = i |-> (the_unity_wrt F) & F .: (((the_inverseOp_wrt F) * T),T) = i |-> (the_unity_wrt F) ) by A2; :: thesis: verum

end;

suppose i <> 0 ; :: thesis: ( F .: (T,((the_inverseOp_wrt F) * T)) = i |-> (the_unity_wrt F) & F .: (((the_inverseOp_wrt F) * T),T) = i |-> (the_unity_wrt F) )

then reconsider C = Seg i as non empty set ;
T is Function of C,D by Lm4;
hence ( F .: (T,((the_inverseOp_wrt F) * T)) = i |-> (the_unity_wrt F) & F .: (((the_inverseOp_wrt F) * T),T) = i |-> (the_unity_wrt F) ) by A1, Th71; :: thesis: verum

end;

end;