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, 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 Nat
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 Nat; :: 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_2:87; :: 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, Th36; :: thesis: verum

end;

end;