let X, S be non empty set ; :: thesis: for R being Relation of X st R is symmetric & field R c= S holds
R is_symmetric_in S
let R be Relation of X; :: thesis: ( R is symmetric & field R c= S implies R is_symmetric_in S )
assume ( R is symmetric & field R c= S ) ; :: thesis: R is_symmetric_in S
then A1: R is_symmetric_in field R by RELAT_2:def 11;
let x, y be set ; :: according to RELAT_2:def 3 :: thesis: ( not x in S or not y in S or not [x,y] in R or [y,x] in R )
assume ( x in S & y in S ) ; :: thesis: ( not [x,y] in R or [y,x] in R )
assume A2: [x,y] in R ; :: thesis: [y,x] in R
then ( x in field R & y in field R ) by RELAT_1:30;
hence [y,x] in R by A1, A2, RELAT_2:def 3; :: thesis: verum