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:
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 :: according to BINOP_1:def 18 :: thesis: verum