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

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

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

then ( T = <*> D & F [:] (T,d1) = <*> D ) by Lm3;
hence F [:] (T,(F . (d1,d2))) = F [:] ((F [:] (T,d1)),d2) ; :: thesis: verum

end;

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

then reconsider C = Seg i as non empty set ;
T is Function of C,D by Lm4;
hence F [:] (T,(F . (d1,d2))) = F [:] ((F [:] (T,d1)),d2) by A1, FUNCOP_1:63; :: thesis: verum

end;

end;