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

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

suppose A2: i = 0 ; :: thesis: F [:] (F [;] d1,T),d2 = F [;] d1,(F [:] T,d2)

then F [;] d1,T = <*> D by Lm2;
then A3: F [:] (F [;] d1,T),d2 = <*> D by FINSEQ_2:99;
F [:] T,d2 = <*> D by A2, Lm3;
hence F [:] (F [;] d1,T),d2 = F [;] d1,(F [:] T,d2) by A3, FINSEQ_2:93; :: thesis: verum

end;

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

then reconsider C = Seg i as non empty set ;
T is Function of C,D by Lm4;
hence F [:] (F [;] d1,T),d2 = F [;] d1,(F [:] T,d2) by A1, FUNCOP_1:74; :: thesis: verum

end;

end;