set X = the carrier of ADG;
set XX = [:the carrier of ADG,the carrier of ADG:];
let P, Q be Relation of [:the carrier of ADG,the carrier of ADG:]; :: thesis: ( ( for a, b, c, d being Element of ADG holds
( [[a,b],[c,d]] in P iff a # d = b # c ) ) & ( for a, b, c, d being Element of ADG holds
( [[a,b],[c,d]] in Q iff a # d = b # c ) ) implies P = Q )
assume that
A5: for a, b, c, d being Element of the carrier of ADG holds
( [[a,b],[c,d]] in P iff a # d = b # c ) and
A6: for a, b, c, d being Element of the carrier of ADG holds
( [[a,b],[c,d]] in Q iff a # d = b # c ) ; :: thesis: P = Q
for x, y being set holds
( [x,y] in P iff [x,y] in Q )

proof

let x, y be set ; :: thesis: ( [x,y] in P iff [x,y] in Q )

A7: now

assume A8: [x,y] in P ; :: thesis: [x,y] in Q
then A9: ( x in [:the carrier of ADG,the carrier of ADG:] & y in [:the carrier of ADG,the carrier of ADG:] ) by ZFMISC_1:106;
then consider a, b being Element of ADG such that
A10: x = [a,b] by DOMAIN_1:9;
consider c, d being Element of ADG such that
A11: y = [c,d] by A9, DOMAIN_1:9;
( [x,y] in P iff a # d = b # c ) by A5, A10, A11;
hence [x,y] in Q by A6, A8, A10, A11; :: thesis: verum

end;

now

assume A12: [x,y] in Q ; :: thesis: [x,y] in P
then A13: ( x in [:the carrier of ADG,the carrier of ADG:] & y in [:the carrier of ADG,the carrier of ADG:] ) by ZFMISC_1:106;
then consider a, b being Element of ADG such that
A14: x = [a,b] by DOMAIN_1:9;
consider c, d being Element of ADG such that
A15: y = [c,d] by A13, DOMAIN_1:9;
( [x,y] in Q iff a # d = b # c ) by A6, A14, A15;
hence [x,y] in P by A5, A12, A14, A15; :: thesis: verum

end;

hence ( [x,y] in P iff [x,y] in Q ) by A7; :: thesis: verum

end;

hence P = Q by RELAT_1:def 2; :: thesis: verum