set N = { [G,f] where G is Element of GroupObjects UN, f is Element of Funcs ([: the carrier of R,1:],1) : verum } ;
let Y1, Y2 be set ; :: thesis: ( ( for y being set 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,1:],1) : verum } & GO x,y,R ) ) ) & ( for y being set 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,1:],1) : verum } & GO x,y,R ) ) ) implies Y1 = Y2 )
assume that
A3: for y being set 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,1:],1) : verum } & GO x,y,R ) ) and
A4: for y being set 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,1:],1) : verum } & GO x,y,R ) ) ; :: thesis: Y1 = Y2

now

let y be set ; :: thesis: ( y in Y1 iff y in Y2 )
( 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,1:],1) : verum } & GO x,y,R ) ) by A3;
hence ( y in Y1 iff y in Y2 ) by A4; :: thesis: verum

end;

hence Y1 = Y2 by TARSKI:1; :: thesis: verum