consider X being set such that
A1: for x being set holds
( x in X iff ( x in field R & S₁[x] ) ) from XBOOLE_0:sch 1();
take X ; :: thesis: for x being set holds
( x in X iff ( x <> a & [x,a] in R ) )
let x be set ; :: thesis: ( x in X iff ( x <> a & [x,a] in R ) )
thus ( x in X implies ( x <> a & [x,a] in R ) ) by A1; :: thesis: ( x <> a & [x,a] in R implies x in X )
assume that
A2: x <> a and
A3: [x,a] in R ; :: thesis: x in X
x in field R by A3, RELAT_1:30;
hence x in X by A1, A2, A3; :: thesis: verum