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, G being BinOp of D st F is_distributive_wrt G holds
F [;] ((G . (d1,d2)),T) = G .: ((F [;] (d1,T)),(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, G being BinOp of D st F is_distributive_wrt G holds
F [;] ((G . (d1,d2)),T) = G .: ((F [;] (d1,T)),(F [;] (d2,T)))
let i be natural Number ; :: thesis: for T being Tuple of i,D
for F, G being BinOp of D st F is_distributive_wrt G holds
F [;] ((G . (d1,d2)),T) = G .: ((F [;] (d1,T)),(F [;] (d2,T)))
let T be Tuple of i,D; :: thesis: for F, G being BinOp of D st F is_distributive_wrt G holds
F [;] ((G . (d1,d2)),T) = G .: ((F [;] (d1,T)),(F [;] (d2,T)))
let F, G be BinOp of D; :: thesis: ( F is_distributive_wrt G implies F [;] ((G . (d1,d2)),T) = G .: ((F [;] (d1,T)),(F [;] (d2,T))) )
assume A1: F is_distributive_wrt G ; :: thesis: F [;] ((G . (d1,d2)),T) = G .: ((F [;] (d1,T)),(F [;] (d2,T)))

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

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

then ( F [;] (d1,T) = <*> D & F [;] ((G . (d1,d2)),T) = <*> D ) by Lm2;
hence F [;] ((G . (d1,d2)),T) = G .: ((F [;] (d1,T)),(F [;] (d2,T))) by FINSEQ_2:73; :: thesis: verum

end;

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

then reconsider C = Seg i as non empty set ;
T is Function of C,D by Lm4;
hence F [;] ((G . (d1,d2)),T) = G .: ((F [;] (d1,T)),(F [;] (d2,T))) by A1, Th35; :: thesis: verum

end;

end;