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;

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 )
;

end;

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;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

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;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