let X be non empty set ; :: thesis: for R being Relation of X st R is_symmetric_in X holds

R is symmetric

let R be Relation of X; :: thesis: ( R is_symmetric_in X implies R is symmetric )

assume A1: R is_symmetric_in X ; :: thesis: R is symmetric

let x, y be object ; :: according to RELAT_2:def 3,RELAT_2:def 11 :: thesis: ( not x in field R or not y in field R or not [x,y] in R or [y,x] in R )

field R c= X \/ X by RELSET_1:8;

hence ( not x in field R or not y in field R or not [x,y] in R or [y,x] in R ) by A1; :: thesis: verum

R is symmetric

let R be Relation of X; :: thesis: ( R is_symmetric_in X implies R is symmetric )

assume A1: R is_symmetric_in X ; :: thesis: R is symmetric

let x, y be object ; :: according to RELAT_2:def 3,RELAT_2:def 11 :: thesis: ( not x in field R or not y in field R or not [x,y] in R or [y,x] in R )

field R c= X \/ X by RELSET_1:8;

hence ( not x in field R or not y in field R or not [x,y] in R or [y,x] in R ) by A1; :: thesis: verum