let D be non empty set ; :: thesis: for F being BinOp of D

for p, q being FinSequence of D st F is commutative & F is associative & F is having_a_unity & len p = len q holds

F . ((F "**" p),(F "**" q)) = F "**" (F .: (p,q))

let F be BinOp of D; :: thesis: for p, q being FinSequence of D st F is commutative & F is associative & F is having_a_unity & len p = len q holds

F . ((F "**" p),(F "**" q)) = F "**" (F .: (p,q))

let p, q be FinSequence of D; :: thesis: ( F is commutative & F is associative & F is having_a_unity & len p = len q implies F . ((F "**" p),(F "**" q)) = F "**" (F .: (p,q)) )

set e = the_unity_wrt F;

assume A1: ( F is commutative & F is associative & F is having_a_unity ) ; :: thesis: ( not len p = len q or F . ((F "**" p),(F "**" q)) = F "**" (F .: (p,q)) )

then ( F . ((the_unity_wrt F),(the_unity_wrt F)) = the_unity_wrt F & ( for d1, d2, d3, d4 being Element of D holds F . ((F . (d1,d2)),(F . (d3,d4))) = F . ((F . (d1,d3)),(F . (d2,d4))) ) ) by Lm3, SETWISEO:15;

hence ( not len p = len q or F . ((F "**" p),(F "**" q)) = F "**" (F .: (p,q)) ) by A1, Th32; :: thesis: verum

for p, q being FinSequence of D st F is commutative & F is associative & F is having_a_unity & len p = len q holds

F . ((F "**" p),(F "**" q)) = F "**" (F .: (p,q))

let F be BinOp of D; :: thesis: for p, q being FinSequence of D st F is commutative & F is associative & F is having_a_unity & len p = len q holds

F . ((F "**" p),(F "**" q)) = F "**" (F .: (p,q))

let p, q be FinSequence of D; :: thesis: ( F is commutative & F is associative & F is having_a_unity & len p = len q implies F . ((F "**" p),(F "**" q)) = F "**" (F .: (p,q)) )

set e = the_unity_wrt F;

assume A1: ( F is commutative & F is associative & F is having_a_unity ) ; :: thesis: ( not len p = len q or F . ((F "**" p),(F "**" q)) = F "**" (F .: (p,q)) )

then ( F . ((the_unity_wrt F),(the_unity_wrt F)) = the_unity_wrt F & ( for d1, d2, d3, d4 being Element of D holds F . ((F . (d1,d2)),(F . (d3,d4))) = F . ((F . (d1,d3)),(F . (d2,d4))) ) ) by Lm3, SETWISEO:15;

hence ( not len p = len q or F . ((F "**" p),(F "**" q)) = F "**" (F .: (p,q)) ) by A1, Th32; :: thesis: verum