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

let d1, d2 be Element of D; :: thesis: for i being natural Number
for T being Tuple of i,D
for F being BinOp of D st F is associative holds
F [;] ((F . (d1,d2)),T) = F [;] (d1,(F [;] (d2,T)))

let i be natural Number ; :: thesis: for T being Tuple of i,D
for F being BinOp of D st F is associative holds
F [;] ((F . (d1,d2)),T) = F [;] (d1,(F [;] (d2,T)))

let T be Tuple of i,D; :: thesis: for F being BinOp of D st F is associative holds
F [;] ((F . (d1,d2)),T) = F [;] (d1,(F [;] (d2,T)))

let F be BinOp of D; :: thesis: ( F is associative implies F [;] ((F . (d1,d2)),T) = F [;] (d1,(F [;] (d2,T))) )
assume A1: F is associative ; :: thesis: F [;] ((F . (d1,d2)),T) = F [;] (d1,(F [;] (d2,T)))
per cases ( i = 0 or i <> 0 ) ;
suppose i = 0 ; :: thesis: F [;] ((F . (d1,d2)),T) = F [;] (d1,(F [;] (d2,T)))
then ( T = <*> D & F [;] (d2,T) = <*> D ) by Lm2;
hence F [;] ((F . (d1,d2)),T) = F [;] (d1,(F [;] (d2,T))) ; :: thesis: verum
end;
suppose i <> 0 ; :: thesis: F [;] ((F . (d1,d2)),T) = F [;] (d1,(F [;] (d2,T)))
then reconsider C = Seg i as non empty set ;
T is Function of C,D by Lm4;
hence F [;] ((F . (d1,d2)),T) = F [;] (d1,(F [;] (d2,T))) by ; :: thesis: verum
end;
end;