let P1, P2 be Relation; :: thesis: ( ( for x, y being set holds
( [x,y] in P1 iff ( y in Y & [x,y] in R ) ) ) & ( for x, y being set holds
( [x,y] in P2 iff ( y in Y & [x,y] in R ) ) ) implies P1 = P2 )
assume that
A2: for x, y being set holds
( [x,y] in P1 iff ( y in Y & [x,y] in R ) ) and
A3: for x, y being set holds
( [x,y] in P2 iff ( y in Y & [x,y] in R ) ) ; :: thesis: P1 = P2

now

let x, y be set ; :: thesis: ( [x,y] in P1 iff [x,y] in P2 )
( [x,y] in P1 iff ( y in Y & [x,y] in R ) ) by A2;
hence ( [x,y] in P1 iff [x,y] in P2 ) by A3; :: thesis: verum

end;

hence P1 = P2 by Def2; :: thesis: verum