let D be non empty set ; :: thesis: for RD being Equivalence_Relation of D
for F being BinOp of D,RD st F is associative holds
F /\/ RD is associative
let RD be Equivalence_Relation of D; :: thesis: for F being BinOp of D,RD st F is associative holds
F /\/ RD is associative
let F be BinOp of D,RD; :: thesis: ( F is associative implies F /\/ RD is associative )
defpred S₁[ Element of Class RD, Element of Class RD, Element of Class RD] means (F /\/ RD) . $1,((F /\/ RD) . $2,$3) = (F /\/ RD) . ((F /\/ RD) . $1,$2),$3;
assume A1: for d, a, b being Element of D holds F . d,(F . a,b) = F . (F . d,a),b ; :: according to BINOP_1:def 14 :: thesis: F /\/ RD is associative

A2: now

let x1, x2, x3 be Element of D; :: thesis: S₁[ EqClass RD,x1, EqClass RD,x2, EqClass RD,x3]
(F /\/ RD) . (EqClass RD,x1),((F /\/ RD) . (EqClass RD,x2),(EqClass RD,x3)) = (F /\/ RD) . (Class RD,x1),(Class RD,(F . x2,x3)) by Th3
.= Class RD,(F . x1,(F . x2,x3)) by Th3
.= Class RD,(F . (F . x1,x2),x3) by A1
.= (F /\/ RD) . (Class RD,(F . x1,x2)),(Class RD,x3) by Th3
.= (F /\/ RD) . ((F /\/ RD) . (EqClass RD,x1),(EqClass RD,x2)),(EqClass RD,x3) by Th3 ;
hence S₁[ EqClass RD,x1, EqClass RD,x2, EqClass RD,x3] ; :: thesis: verum

end;

thus for c1, c2, c3 being Element of Class RD holds S₁[c1,c2,c3] from FILTER_1:sch 3(A2); :: according to BINOP_1:def 14 :: thesis: verum