let D be non empty set ; :: thesis: for d being Element of D
for i being natural Number
for T1, T2 being Tuple of i,D
for F being BinOp of D st F is associative holds
F .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2)))

let d be Element of D; :: thesis: for i being natural Number
for T1, T2 being Tuple of i,D
for F being BinOp of D st F is associative holds
F .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2)))

let i be natural Number ; :: thesis: for T1, T2 being Tuple of i,D
for F being BinOp of D st F is associative holds
F .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2)))

let T1, T2 be Tuple of i,D; :: thesis: for F being BinOp of D st F is associative holds
F .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2)))

let F be BinOp of D; :: thesis: ( F is associative implies F .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2))) )
assume A1: F is associative ; :: thesis: F .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2)))
per cases ( i = 0 or i <> 0 ) ;
suppose A2: i = 0 ; :: thesis: F .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2)))
then F [:] (T1,d) = <*> D by Lm3;
then A3: F .: ((F [:] (T1,d)),T2) = <*> D by FINSEQ_2:73;
F [;] (d,T2) = <*> D by ;
hence F .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2))) by ; :: thesis: verum
end;
suppose i <> 0 ; :: thesis: F .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2)))
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 .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2))) by ; :: thesis: verum
end;
end;