let D be non empty set ; :: thesis: for d being Element of D
for i being Nat
for T being Tuple of ,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 Nat
for T being Tuple of ,D
for F being BinOp of D st F is commutative holds
F [;] d,T = F [:] T,d
let i be Nat; :: thesis: for T being Tuple of ,D
for F being BinOp of D st F is commutative holds
F [;] d,T = F [:] T,d
let T be Tuple of ,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 A2, Lm3; :: 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 A1, FUNCOP_1:79; :: thesis: verum

end;

end;