let D be non empty set ; :: thesis: for i being natural Number

for T1, T2, T3 being Tuple of i,D

for F being BinOp of D st F is associative holds

F .: ((F .: (T1,T2)),T3) = F .: (T1,(F .: (T2,T3)))

let i be natural Number ; :: thesis: for T1, T2, T3 being Tuple of i,D

for F being BinOp of D st F is associative holds

F .: ((F .: (T1,T2)),T3) = F .: (T1,(F .: (T2,T3)))

let T1, T2, T3 be Tuple of i,D; :: thesis: for F being BinOp of D st F is associative holds

F .: ((F .: (T1,T2)),T3) = F .: (T1,(F .: (T2,T3)))

let F be BinOp of D; :: thesis: ( F is associative implies F .: ((F .: (T1,T2)),T3) = F .: (T1,(F .: (T2,T3))) )

assume A1: F is associative ; :: thesis: F .: ((F .: (T1,T2)),T3) = F .: (T1,(F .: (T2,T3)))

for T1, T2, T3 being Tuple of i,D

for F being BinOp of D st F is associative holds

F .: ((F .: (T1,T2)),T3) = F .: (T1,(F .: (T2,T3)))

let i be natural Number ; :: thesis: for T1, T2, T3 being Tuple of i,D

for F being BinOp of D st F is associative holds

F .: ((F .: (T1,T2)),T3) = F .: (T1,(F .: (T2,T3)))

let T1, T2, T3 be Tuple of i,D; :: thesis: for F being BinOp of D st F is associative holds

F .: ((F .: (T1,T2)),T3) = F .: (T1,(F .: (T2,T3)))

let F be BinOp of D; :: thesis: ( F is associative implies F .: ((F .: (T1,T2)),T3) = F .: (T1,(F .: (T2,T3))) )

assume A1: F is associative ; :: thesis: F .: ((F .: (T1,T2)),T3) = F .: (T1,(F .: (T2,T3)))

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

end;

suppose A2:
i = 0
; :: thesis: F .: ((F .: (T1,T2)),T3) = F .: (T1,(F .: (T2,T3)))

then
F .: (T1,T2) = <*> D
by Lm1;

then A3: F .: ((F .: (T1,T2)),T3) = <*> D by FINSEQ_2:73;

F .: (T2,T3) = <*> D by A2, Lm1;

hence F .: ((F .: (T1,T2)),T3) = F .: (T1,(F .: (T2,T3))) by A3, FINSEQ_2:73; :: thesis: verum

end;then A3: F .: ((F .: (T1,T2)),T3) = <*> D by FINSEQ_2:73;

F .: (T2,T3) = <*> D by A2, Lm1;

hence F .: ((F .: (T1,T2)),T3) = F .: (T1,(F .: (T2,T3))) by A3, FINSEQ_2:73; :: thesis: verum

suppose
i <> 0
; :: thesis: F .: ((F .: (T1,T2)),T3) = F .: (T1,(F .: (T2,T3)))

then reconsider C = Seg i as non empty set ;

A4: T3 is Function of C,D by Lm4;

( T1 is Function of C,D & T2 is Function of C,D ) by Lm4;

hence F .: ((F .: (T1,T2)),T3) = F .: (T1,(F .: (T2,T3))) by A1, A4, FUNCOP_1:61; :: thesis: verum

end;A4: T3 is Function of C,D by Lm4;

( T1 is Function of C,D & T2 is Function of C,D ) by Lm4;

hence F .: ((F .: (T1,T2)),T3) = F .: (T1,(F .: (T2,T3))) by A1, A4, FUNCOP_1:61; :: thesis: verum