let X, Y be set ; :: thesis: for R being Relation holds (Y |` R) | X = Y |` (R | X)
let R be Relation; :: thesis: (Y |` R) | X = Y |` (R | X)
let x be object ; :: according to RELAT_1:def 2 :: thesis: for b being object holds
( [x,b] in (Y |` R) | X iff [x,b] in Y |` (R | X) )
let y be object ; :: thesis: ( [x,y] in (Y |` R) | X iff [x,y] in Y |` (R | X) )
A1: ( ( [x,y] in R & x in X ) iff [x,y] in R | X ) by Def9;
( [x,y] in Y |` R iff ( [x,y] in R & y in Y ) ) by Def10;
hence ( [x,y] in (Y |` R) | X iff [x,y] in Y |` (R | X) ) by A1, Def9, Def10; :: thesis: verum