set N = { [G,f] where G is Element of GroupObjects UN, f is Element of Funcs ([: the carrier of R, the carrier of G:], the carrier of G) : verum } ;
let Y1, Y2 be set ; :: thesis:
assume that
A3: for y being object holds
( y in Y1 iff ex x being set st
( x in { [G,f] where G is Element of GroupObjects UN, f is Element of Funcs ([: the carrier of R, the carrier of G:], the carrier of G) : verum } & GO x,y,R ) ) and
A4: for y being object holds
( y in Y2 iff ex x being set st
( x in { [G,f] where G is Element of GroupObjects UN, f is Element of Funcs ([: the carrier of R, the carrier of G:], the carrier of G) : verum } & GO x,y,R ) ) ; :: thesis:

let y be object ; :: thesis:
( y in Y1 iff ex x being set st
( x in { [G,f] where G is Element of GroupObjects UN, f is Element of Funcs ([: the carrier of R, the carrier of G:], the carrier of G) : verum } & GO x,y,R ) ) by A3;
hence ( y in Y1 iff y in Y2 ) by A4; :: thesis:

end;