let D be non empty set ; :: thesis: for d1, d2 being Element of D
for i being Nat
for T being Element of i -tuples_on 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 Nat
for T being Element of i -tuples_on 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 Nat; :: thesis: for T being Element of i -tuples_on 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 Element of i -tuples_on 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 Lm4, FINSEQ_2:113;
then ( F [:] T,(F . d1,d2) = <*> D & F [:] (F [:] T,d1),d2 = <*> D ) ;
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 Lm5;
hence F [:] T,(F . d1,d2) = F [:] (F [:] T,d1),d2 by A1, FUNCOP_1:78; :: thesis: verum

end;

end;