let D be non empty set ; :: thesis: for d1, d2 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 associative holds
F [:] ((F [;] (d1,T)),d2) = F [;] (d1,(F [:] (T,d2)))
let d1, d2 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 associative holds
F [:] ((F [;] (d1,T)),d2) = F [;] (d1,(F [:] (T,d2)))
let i be natural Number ; :: thesis: for T being Tuple of i,D
for F being BinOp of D st F is associative holds
F [:] ((F [;] (d1,T)),d2) = F [;] (d1,(F [:] (T,d2)))
let T be Tuple of i,D; :: thesis: for F being BinOp of D st F is associative holds
F [:] ((F [;] (d1,T)),d2) = F [;] (d1,(F [:] (T,d2)))
let F be BinOp of D; :: thesis: ( F is associative implies F [:] ((F [;] (d1,T)),d2) = F [;] (d1,(F [:] (T,d2))) )
assume A1: F is associative ; :: thesis: F [:] ((F [;] (d1,T)),d2) = F [;] (d1,(F [:] (T,d2)))

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

suppose A2: i = 0 ; :: thesis: F [:] ((F [;] (d1,T)),d2) = F [;] (d1,(F [:] (T,d2)))

then F [;] (d1,T) = <*> D by Lm2;
then A3: F [:] ((F [;] (d1,T)),d2) = <*> D by FINSEQ_2:85;
F [:] (T,d2) = <*> D by A2, Lm3;
hence F [:] ((F [;] (d1,T)),d2) = F [;] (d1,(F [:] (T,d2))) by A3, FINSEQ_2:79; :: thesis: verum

end;

suppose i <> 0 ; :: thesis: F [:] ((F [;] (d1,T)),d2) = F [;] (d1,(F [:] (T,d2)))

then reconsider C = Seg i as non empty set ;
T is Function of C,D by Lm4;
hence F [:] ((F [;] (d1,T)),d2) = F [;] (d1,(F [:] (T,d2))) by A1, FUNCOP_1:59; :: thesis: verum

end;

end;