let D be non empty set ; :: thesis: for i being Nat
for T1, T2 being Tuple of i,D
for F being BinOp of D st F is commutative holds
F .: T1,T2 = F .: T2,T1
let i be Nat; :: thesis: for T1, T2 being Tuple of i,D
for F being BinOp of D st F is commutative holds
F .: T1,T2 = F .: T2,T1
let T1, T2 be Tuple of i,D; :: thesis: for F being BinOp of D st F is commutative holds
F .: T1,T2 = F .: T2,T1
let F be BinOp of D; :: thesis: ( F is commutative implies F .: T1,T2 = F .: T2,T1 )
assume A1: F is commutative ; :: thesis: F .: T1,T2 = F .: T2,T1

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

suppose A2: i = 0 ; :: thesis: F .: T1,T2 = F .: T2,T1

then F .: T1,T2 = <*> D by Lm1;
hence F .: T1,T2 = F .: T2,T1 by A2, Lm1; :: thesis: verum

end;

suppose i <> 0 ; :: thesis: F .: T1,T2 = F .: T2,T1

then reconsider C = Seg i as non empty set ;
( T1 is Function of C,D & T2 is Function of C,D ) by Lm4;
hence F .: T1,T2 = F .: T2,T1 by A1, FUNCOP_1:80; :: thesis: verum

end;

end;