let D be non empty set ; :: thesis: for d being Element of D
for i being Nat
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 Nat
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 Nat; :: 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_2:87;
F [;] d,T2 = <*> D by A2, Lm2;
hence F .: (F [:] T1,d),T2 = F .: T1,(F [;] d,T2) by A3, FINSEQ_2:87; :: 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 A1, FUNCOP_1:75; :: thesis: verum

end;

end;