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 set ; :: 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 )
A2: field R c= X \/ X by RELSET_1:19;
assume A3: x in field R ; :: thesis: ( not y in field R or not [x,y] in R or [y,x] in R )
assume A4: y in field R ; :: thesis: ( not [x,y] in R or [y,x] in R )
assume [x,y] in R ; :: thesis: [y,x] in R
hence [y,x] in R by A1, A2, A3, A4, RELAT_2:def 3; :: thesis: verum