let D be non empty set ; :: thesis: for d being Element of D
for i being Nat
for T1, T2 being Tuple of ,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 Tuple of ,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 Tuple of ,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 Tuple of ,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 A2: i = 0 ; :: thesis: (F [:] (id D),d) * (F .: T1,T2) = F .: T1,((F [:] (id D),d) * T2)

then F .: T1,T2 = <*> D by Lm1;
then A3: (F [:] (id D),d) * (F .: T1,T2) = <*> D ;
T1 = <*> D by A2;
hence (F [:] (id D),d) * (F .: T1,T2) = F .: T1,((F [:] (id D),d) * T2) by A3, FINSEQ_2:87; :: 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 Lm4;
hence (F [:] (id D),d) * (F .: T1,T2) = F .: T1,((F [:] (id D),d) * T2) by A1, Th26; :: thesis: verum

end;

end;