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 [:] (T,(G . (d1,d2))) = G .: ((F [:] (T,d1)),(F [:] (T,d2)))

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 [:] (T,(G . (d1,d2))) = G .: ((F [:] (T,d1)),(F [:] (T,d2)))

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 [:] (T,(G . (d1,d2))) = G .: ((F [:] (T,d1)),(F [:] (T,d2)))

let T be Tuple of i,D; :: thesis: for F, G being BinOp of D st F is_distributive_wrt G holds

F [:] (T,(G . (d1,d2))) = G .: ((F [:] (T,d1)),(F [:] (T,d2)))

let F, G be BinOp of D; :: thesis: ( F is_distributive_wrt G implies F [:] (T,(G . (d1,d2))) = G .: ((F [:] (T,d1)),(F [:] (T,d2))) )

assume A1: F is_distributive_wrt G ; :: thesis: F [:] (T,(G . (d1,d2))) = G .: ((F [:] (T,d1)),(F [:] (T,d2)))

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 [:] (T,(G . (d1,d2))) = G .: ((F [:] (T,d1)),(F [:] (T,d2)))

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 [:] (T,(G . (d1,d2))) = G .: ((F [:] (T,d1)),(F [:] (T,d2)))

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 [:] (T,(G . (d1,d2))) = G .: ((F [:] (T,d1)),(F [:] (T,d2)))

let T be Tuple of i,D; :: thesis: for F, G being BinOp of D st F is_distributive_wrt G holds

F [:] (T,(G . (d1,d2))) = G .: ((F [:] (T,d1)),(F [:] (T,d2)))

let F, G be BinOp of D; :: thesis: ( F is_distributive_wrt G implies F [:] (T,(G . (d1,d2))) = G .: ((F [:] (T,d1)),(F [:] (T,d2))) )

assume A1: F is_distributive_wrt G ; :: thesis: F [:] (T,(G . (d1,d2))) = G .: ((F [:] (T,d1)),(F [:] (T,d2)))

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

end;