let D be non empty set ; for e being Element of D
for F, G being BinOp of D
for p, q being FinSequence of D st F is commutative & F is associative & F is having_a_unity & e = the_unity_wrt F & G . (e,e) = e & ( for d1, d2, d3, d4 being Element of D holds F . ((G . (d1,d2)),(G . (d3,d4))) = G . ((F . (d1,d3)),(F . (d2,d4))) ) & len p = len q holds
G . ((F "**" p),(F "**" q)) = F "**" (G .: (p,q))
let e be Element of D; for F, G being BinOp of D
for p, q being FinSequence of D st F is commutative & F is associative & F is having_a_unity & e = the_unity_wrt F & G . (e,e) = e & ( for d1, d2, d3, d4 being Element of D holds F . ((G . (d1,d2)),(G . (d3,d4))) = G . ((F . (d1,d3)),(F . (d2,d4))) ) & len p = len q holds
G . ((F "**" p),(F "**" q)) = F "**" (G .: (p,q))
let F, G be BinOp of D; for p, q being FinSequence of D st F is commutative & F is associative & F is having_a_unity & e = the_unity_wrt F & G . (e,e) = e & ( for d1, d2, d3, d4 being Element of D holds F . ((G . (d1,d2)),(G . (d3,d4))) = G . ((F . (d1,d3)),(F . (d2,d4))) ) & len p = len q holds
G . ((F "**" p),(F "**" q)) = F "**" (G .: (p,q))
let p, q be FinSequence of D; ( F is commutative & F is associative & F is having_a_unity & e = the_unity_wrt F & G . (e,e) = e & ( for d1, d2, d3, d4 being Element of D holds F . ((G . (d1,d2)),(G . (d3,d4))) = G . ((F . (d1,d3)),(F . (d2,d4))) ) & len p = len q implies G . ((F "**" p),(F "**" q)) = F "**" (G .: (p,q)) )
assume that
A1:
( F is commutative & F is associative & F is having_a_unity & e = the_unity_wrt F )
and
A2:
G . (e,e) = e
and
A3:
for d1, d2, d3, d4 being Element of D holds F . ((G . (d1,d2)),(G . (d3,d4))) = G . ((F . (d1,d3)),(F . (d2,d4)))
and
A4:
len p = len q
; G . ((F "**" p),(F "**" q)) = F "**" (G .: (p,q))
A5:
len p = len (G .: (p,q))
by A4, FINSEQ_2:72;
A6:
dom (G .: (p,q)) = Seg (len (G .: (p,q)))
by FINSEQ_1:def 3;
A7:
dom q = Seg (len q)
by FINSEQ_1:def 3;
A8:
dom p = Seg (len p)
by FINSEQ_1:def 3;
thus G . ((F "**" p),(F "**" q)) =
G . ((F $$ ((findom p),([#] (p,e)))),(F "**" q))
by A1, Def2
.=
G . ((F $$ ((findom p),([#] (p,e)))),(F $$ ((findom q),([#] (q,e)))))
by A1, Def2
.=
F $$ ((findom p),(G .: (([#] (p,e)),([#] (q,e)))))
by A1, A2, A3, A4, A8, A7, Th9
.=
F $$ ((findom (G .: (p,q))),([#] ((G .: (p,q)),e)))
by A2, A4, A5, A8, A6, Lm4
.=
F "**" (G .: (p,q))
by A1, Def2
; verum