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_left_distributive_wrt G holds
F /\/ RD is_left_distributive_wrt G /\/ RD
let RD be Equivalence_Relation of D; :: thesis: for F, G being BinOp of D,RD st F is_left_distributive_wrt G holds
F /\/ RD is_left_distributive_wrt G /\/ RD
let F, G be BinOp of D,RD; :: thesis: ( F is_left_distributive_wrt G implies F /\/ RD is_left_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) . ($1,((G /\/ RD) . ($2,$3))) = (G /\/ RD) . (((F /\/ RD) . ($1,$2)),((F /\/ RD) . ($1,$3)));
assume A1: for d, a, b being Element of D holds F . (d,(G . (a,b))) = G . ((F . (d,a)),(F . (d,b))) ; :: according to BINOP_1:def 18 :: thesis: F /\/ RD is_left_distributive_wrt G /\/ RD

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

let x1, x2, x3 be Element of D; :: thesis: S₁[ EqClass (RD,x1), EqClass (RD,x2), EqClass (RD,x3)]
(F /\/ RD) . (H₃(x1),((G /\/ RD) . (H₃(x2),H₃(x3)))) = (F /\/ RD) . (H₃(x1),H₃(G . (x2,x3))) by Th3
.= H₃(F . (x1,(G . (x2,x3)))) by Th3
.= H₃(G . ((F . (x1,x2)),(F . (x1,x3)))) by A1
.= (G /\/ RD) . (H₃(F . (x1,x2)),H₃(F . (x1,x3))) by Th3
.= (G /\/ RD) . (((F /\/ RD) . (H₃(x1),H₃(x2))),H₃(F . (x1,x3))) by Th3
.= (G /\/ RD) . (((F /\/ RD) . (H₃(x1),H₃(x2))),((F /\/ RD) . (H₃(x1),H₃(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 18 :: thesis: verum