let D be non empty set ; :: thesis: for i being Nat
for T being Element of i -tuples_on 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 Nat; :: thesis: for T being Element of i -tuples_on 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 Element of i -tuples_on 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) )
set e = the_unity_wrt F;
reconsider uT = (the_inverseOp_wrt F) * T as Element of i -tuples_on D by FINSEQ_2:133;

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

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 ( F .: T,uT = <*> D & F .: uT,T = <*> D & i |-> (the_unity_wrt F) = <*> D ) by Lm2;
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) ) ; :: 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 & i |-> (the_unity_wrt F) = C --> (the_unity_wrt F) ) by Lm5;
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, Th75; :: thesis: verum

end;

end;