let D be non empty set ; :: thesis: for RD being Equivalence_Relation of D
for F, G being BinOp of D,RD st F is_right_distributive_wrt G holds
F /\/ RD is_right_distributive_wrt G /\/ RD
let RD be Equivalence_Relation of D; :: thesis: for F, G being BinOp of D,RD st F is_right_distributive_wrt G holds
F /\/ RD is_right_distributive_wrt G /\/ RD
let F, G be BinOp of D,RD; :: thesis: ( F is_right_distributive_wrt G implies F /\/ RD is_right_distributive_wrt G /\/ RD )
deffunc H₃( Element of D) -> Element of Class RD = EqClass (RD,$1);
defpred S₁[ Element of Class RD, Element of Class RD, Element of Class RD] means (F /\/ RD) . (((G /\/ RD) . ($1,$2)),$3) = (G /\/ RD) . (((F /\/ RD) . ($1,$3)),((F /\/ RD) . ($2,$3)));
assume A1: for a, b, d being Element of D holds F . ((G . (a,b)),d) = G . ((F . (a,d)),(F . (b,d))) ; :: according to BINOP_1:def 19 :: thesis: F /\/ RD is_right_distributive_wrt G /\/ RD

A2: now :: thesis: for x2, x3, x1 being Element of D holds S₁[ EqClass (RD,x2), EqClass (RD,x3), EqClass (RD,x1)]

let x2, x3, x1 be Element of D; :: thesis: S₁[ EqClass (RD,x2), EqClass (RD,x3), EqClass (RD,x1)]
(F /\/ RD) . (((G /\/ RD) . (H₃(x2),H₃(x3))),H₃(x1)) = (F /\/ RD) . (H₃(G . (x2,x3)),H₃(x1)) by Th3
.= H₃(F . ((G . (x2,x3)),x1)) by Th3
.= H₃(G . ((F . (x2,x1)),(F . (x3,x1)))) by A1
.= (G /\/ RD) . (H₃(F . (x2,x1)),H₃(F . (x3,x1))) by Th3
.= (G /\/ RD) . (((F /\/ RD) . (H₃(x2),H₃(x1))),H₃(F . (x3,x1))) by Th3
.= (G /\/ RD) . (((F /\/ RD) . (H₃(x2),H₃(x1))),((F /\/ RD) . (H₃(x3),H₃(x1)))) by Th3 ;
hence S₁[ EqClass (RD,x2), EqClass (RD,x3), EqClass (RD,x1)] ; :: thesis: verum

end;

thus for c2, c3, c1 being Element of Class RD holds S₁[c2,c3,c1] from FILTER_1:sch 3(A2); :: according to BINOP_1:def 19 :: thesis: verum