let D be non empty set ; :: thesis: for d being Element of D
for i being natural Number
for T being Tuple of i,D
for F being BinOp of D st F is commutative holds
F [;] (d,T) = F [:] (T,d)

let d be Element of D; :: thesis: for i being natural Number
for T being Tuple of i,D
for F being BinOp of D st F is commutative holds
F [;] (d,T) = F [:] (T,d)

let i be natural Number ; :: thesis: for T being Tuple of i,D
for F being BinOp of D st F is commutative holds
F [;] (d,T) = F [:] (T,d)

let T be Tuple of i,D; :: thesis: for F being BinOp of D st F is commutative holds
F [;] (d,T) = F [:] (T,d)

let F be BinOp of D; :: thesis: ( F is commutative implies F [;] (d,T) = F [:] (T,d) )
assume A1: F is commutative ; :: thesis: F [;] (d,T) = F [:] (T,d)
per cases ( i = 0 or i <> 0 ) ;
suppose A2: i = 0 ; :: thesis: F [;] (d,T) = F [:] (T,d)
then F [;] (d,T) = <*> D by Lm2;
hence F [;] (d,T) = F [:] (T,d) by ; :: thesis: verum
end;
suppose i <> 0 ; :: thesis: F [;] (d,T) = F [:] (T,d)
then reconsider C = Seg i as non empty set ;
T is Function of C,D by Lm4;
hence F [;] (d,T) = F [:] (T,d) by ; :: thesis: verum
end;
end;