let D be non empty set ; :: thesis: for i being natural Number
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 natural Number ; :: 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 ; :: 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 ; :: thesis: verum
end;
end;