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 H_{3}( Element of D) -> Element of Class RD = EqClass (RD,$1);

defpred S_{1}[ 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

_{1}[c1,c2,c3]
from FILTER_1:sch 3(A2); :: according to BINOP_1:def 18 :: thesis: verum

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 H

defpred S

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 S_{1}[ EqClass (RD,x1), EqClass (RD,x2), EqClass (RD,x3)]

thus
for c1, c2, c3 being Element of Class RD holds Slet x1, x2, x3 be Element of D; :: thesis: S_{1}[ EqClass (RD,x1), EqClass (RD,x2), EqClass (RD,x3)]

(F /\/ RD) . (H_{3}(x1),((G /\/ RD) . (H_{3}(x2),H_{3}(x3)))) =
(F /\/ RD) . (H_{3}(x1),H_{3}(G . (x2,x3)))
by Th3

.= H_{3}(F . (x1,(G . (x2,x3))))
by Th3

.= H_{3}(G . ((F . (x1,x2)),(F . (x1,x3))))
by A1

.= (G /\/ RD) . (H_{3}(F . (x1,x2)),H_{3}(F . (x1,x3)))
by Th3

.= (G /\/ RD) . (((F /\/ RD) . (H_{3}(x1),H_{3}(x2))),H_{3}(F . (x1,x3)))
by Th3

.= (G /\/ RD) . (((F /\/ RD) . (H_{3}(x1),H_{3}(x2))),((F /\/ RD) . (H_{3}(x1),H_{3}(x3))))
by Th3
;

hence S_{1}[ EqClass (RD,x1), EqClass (RD,x2), EqClass (RD,x3)]
; :: thesis: verum

end;(F /\/ RD) . (H

.= H

.= H

.= (G /\/ RD) . (H

.= (G /\/ RD) . (((F /\/ RD) . (H

.= (G /\/ RD) . (((F /\/ RD) . (H

hence S