let D be non empty set ; :: thesis: for RD being Equivalence_Relation of D

for F being BinOp of D,RD st F is commutative holds

F /\/ RD is commutative

let RD be Equivalence_Relation of D; :: thesis: for F being BinOp of D,RD st F is commutative holds

F /\/ RD is commutative

let F be BinOp of D,RD; :: thesis: ( F is commutative implies F /\/ RD is commutative )

defpred S_{1}[ Element of Class RD, Element of Class RD] means (F /\/ RD) . ($1,$2) = (F /\/ RD) . ($2,$1);

assume A1: for a, b being Element of D holds F . (a,b) = F . (b,a) ; :: according to BINOP_1:def 2 :: thesis: F /\/ RD is commutative

_{1}[c1,c2]
from FILTER_1:sch 2(A2); :: according to BINOP_1:def 2 :: thesis: verum

for F being BinOp of D,RD st F is commutative holds

F /\/ RD is commutative

let RD be Equivalence_Relation of D; :: thesis: for F being BinOp of D,RD st F is commutative holds

F /\/ RD is commutative

let F be BinOp of D,RD; :: thesis: ( F is commutative implies F /\/ RD is commutative )

defpred S

assume A1: for a, b being Element of D holds F . (a,b) = F . (b,a) ; :: according to BINOP_1:def 2 :: thesis: F /\/ RD is commutative

A2: now :: thesis: for x1, x2 being Element of D holds S_{1}[ EqClass (RD,x1), EqClass (RD,x2)]

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

(F /\/ RD) . ((EqClass (RD,x1)),(EqClass (RD,x2))) = Class (RD,(F . (x1,x2))) by Th3

.= Class (RD,(F . (x2,x1))) by A1

.= (F /\/ RD) . ((EqClass (RD,x2)),(EqClass (RD,x1))) by Th3 ;

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

end;(F /\/ RD) . ((EqClass (RD,x1)),(EqClass (RD,x2))) = Class (RD,(F . (x1,x2))) by Th3

.= Class (RD,(F . (x2,x1))) by A1

.= (F /\/ RD) . ((EqClass (RD,x2)),(EqClass (RD,x1))) by Th3 ;

hence S