let D be non empty set ; :: thesis: for i being Nat
for T1, T2, T3 being Tuple of ,D
for F being BinOp of D st F is associative holds
F .: (F .: T1,T2),T3 = F .: T1,(F .: T2,T3)
let i be Nat; :: thesis: for T1, T2, T3 being Tuple of ,D
for F being BinOp of D st F is associative holds
F .: (F .: T1,T2),T3 = F .: T1,(F .: T2,T3)
let T1, T2, T3 be Tuple of ,D; :: thesis: for F being BinOp of D st F is associative holds
F .: (F .: T1,T2),T3 = F .: T1,(F .: T2,T3)
let F be BinOp of D; :: thesis: ( F is associative implies F .: (F .: T1,T2),T3 = F .: T1,(F .: T2,T3) )
assume A1: F is associative ; :: thesis: F .: (F .: T1,T2),T3 = F .: T1,(F .: T2,T3)

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

suppose A2: i = 0 ; :: thesis: F .: (F .: T1,T2),T3 = F .: T1,(F .: T2,T3)

then F .: T1,T2 = <*> D by Lm1;
then A3: F .: (F .: T1,T2),T3 = <*> D by FINSEQ_2:87;
F .: T2,T3 = <*> D by A2, Lm1;
hence F .: (F .: T1,T2),T3 = F .: T1,(F .: T2,T3) by A3, FINSEQ_2:87; :: thesis: verum

end;

suppose i <> 0 ; :: thesis: F .: (F .: T1,T2),T3 = F .: T1,(F .: T2,T3)

then reconsider C = Seg i as non empty set ;
A4: T3 is Function of C,D by Lm4;
( T1 is Function of C,D & T2 is Function of C,D ) by Lm4;
hence F .: (F .: T1,T2),T3 = F .: T1,(F .: T2,T3) by A1, A4, FUNCOP_1:76; :: thesis: verum

end;

end;