let D be non empty set ; :: thesis: for e being Element of D
for i being Nat
for T being Element of i -tuples_on D
for F, G being BinOp of D st F is associative & F is having_a_unity & e = the_unity_wrt F & F is having_an_inverseOp & G is_distributive_wrt F holds
G [;] e,T = i |-> e
let e be Element of D; :: thesis: for i being Nat
for T being Element of i -tuples_on D
for F, G being BinOp of D st F is associative & F is having_a_unity & e = the_unity_wrt F & F is having_an_inverseOp & G is_distributive_wrt F holds
G [;] e,T = i |-> e
let i be Nat; :: thesis: for T being Element of i -tuples_on D
for F, G being BinOp of D st F is associative & F is having_a_unity & e = the_unity_wrt F & F is having_an_inverseOp & G is_distributive_wrt F holds
G [;] e,T = i |-> e
let T be Element of i -tuples_on D; :: thesis: for F, G being BinOp of D st F is associative & F is having_a_unity & e = the_unity_wrt F & F is having_an_inverseOp & G is_distributive_wrt F holds
G [;] e,T = i |-> e
let F, G be BinOp of D; :: thesis: ( F is associative & F is having_a_unity & e = the_unity_wrt F & F is having_an_inverseOp & G is_distributive_wrt F implies G [;] e,T = i |-> e )
assume A1: ( F is associative & F is having_a_unity & e = the_unity_wrt F & F is having_an_inverseOp & G is_distributive_wrt F ) ; :: thesis: G [;] e,T = i |-> e

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

suppose i = 0 ; :: thesis: G [;] e,T = i |-> e

then ( G [;] e,T = <*> D & i |-> e = <*> D ) by Lm3;
hence G [;] e,T = i |-> e ; :: thesis: verum

end;

suppose i <> 0 ; :: thesis: G [;] e,T = i |-> e

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 G [;] e,T = i |-> e by A1, Th79; :: thesis: verum

end;

end;