let D be non empty set ; :: thesis: for d being Element of D
for i being Nat
for T1, T2 being Element of i -tuples_on D
for F being BinOp of D st F is associative holds
(F [:] (id D),d) * (F .: T1,T2) = F .: T1,((F [:] (id D),d) * T2)
let d be Element of D; :: thesis: for i being Nat
for T1, T2 being Element of i -tuples_on D
for F being BinOp of D st F is associative holds
(F [:] (id D),d) * (F .: T1,T2) = F .: T1,((F [:] (id D),d) * T2)
let i be Nat; :: thesis: for T1, T2 being Element of i -tuples_on D
for F being BinOp of D st F is associative holds
(F [:] (id D),d) * (F .: T1,T2) = F .: T1,((F [:] (id D),d) * T2)
let T1, T2 be Element of i -tuples_on D; :: thesis: for F being BinOp of D st F is associative holds
(F [:] (id D),d) * (F .: T1,T2) = F .: T1,((F [:] (id D),d) * T2)
let F be BinOp of D; :: thesis: ( F is associative implies (F [:] (id D),d) * (F .: T1,T2) = F .: T1,((F [:] (id D),d) * T2) )
assume A1: F is associative ; :: thesis: (F [:] (id D),d) * (F .: T1,T2) = F .: T1,((F [:] (id D),d) * T2)

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

suppose i = 0 ; :: thesis: (F [:] (id D),d) * (F .: T1,T2) = F .: T1,((F [:] (id D),d) * T2)

then ( F .: T1,T2 = <*> D & T1 = <*> D ) by Lm2, FINSEQ_2:113;
then ( (F [:] (id D),d) * (F .: T1,T2) = <*> D & F .: T1,((F [:] (id D),d) * T2) = <*> D ) by FINSEQ_2:87;
hence (F [:] (id D),d) * (F .: T1,T2) = F .: T1,((F [:] (id D),d) * T2) ; :: thesis: verum

end;

suppose i <> 0 ; :: thesis: (F [:] (id D),d) * (F .: T1,T2) = F .: T1,((F [:] (id D),d) * T2)

then reconsider C = Seg i as non empty set ;
( T1 is Function of C,D & T2 is Function of C,D ) by Lm5;
hence (F [:] (id D),d) * (F .: T1,T2) = F .: T1,((F [:] (id D),d) * T2) by A1, Th26; :: thesis: verum

end;

end;