let D be non empty set ; :: thesis: for d being Element of D

for i being natural Number

for T1, T2 being Tuple of i,D

for F being BinOp of D st F is associative holds

F .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2)))

let d be Element of D; :: thesis: for i being natural Number

for T1, T2 being Tuple of i,D

for F being BinOp of D st F is associative holds

F .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2)))

let i be natural Number ; :: thesis: for T1, T2 being Tuple of i,D

for F being BinOp of D st F is associative holds

F .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2)))

let T1, T2 be Tuple of i,D; :: thesis: for F being BinOp of D st F is associative holds

F .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2)))

let F be BinOp of D; :: thesis: ( F is associative implies F .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2))) )

assume A1: F is associative ; :: thesis: F .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2)))

for i being natural Number

for T1, T2 being Tuple of i,D

for F being BinOp of D st F is associative holds

F .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2)))

let d be Element of D; :: thesis: for i being natural Number

for T1, T2 being Tuple of i,D

for F being BinOp of D st F is associative holds

F .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2)))

let i be natural Number ; :: thesis: for T1, T2 being Tuple of i,D

for F being BinOp of D st F is associative holds

F .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2)))

let T1, T2 be Tuple of i,D; :: thesis: for F being BinOp of D st F is associative holds

F .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2)))

let F be BinOp of D; :: thesis: ( F is associative implies F .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2))) )

assume A1: F is associative ; :: thesis: F .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2)))

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

end;

suppose A2:
i = 0
; :: thesis: F .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2)))

then
F [:] (T1,d) = <*> D
by Lm3;

then A3: F .: ((F [:] (T1,d)),T2) = <*> D by FINSEQ_2:73;

F [;] (d,T2) = <*> D by A2, Lm2;

hence F .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2))) by A3, FINSEQ_2:73; :: thesis: verum

end;then A3: F .: ((F [:] (T1,d)),T2) = <*> D by FINSEQ_2:73;

F [;] (d,T2) = <*> D by A2, Lm2;

hence F .: ((F [:] (T1,d)),T2) = F .: (T1,(F [;] (d,T2))) by A3, FINSEQ_2:73; :: thesis: verum