let f1, f2 be Relation of [:(Seg n),(Seg m):]; :: thesis: ( ( for x being set st x in [:(Seg n),(Seg m):] holds
for i, j being Element of NAT st x = [i,j] holds
Im f1,x = [:(Im (Nbdl1 n),i),(Im (Nbdl1 m),j):] ) & ( for x being set st x in [:(Seg n),(Seg m):] holds
for i, j being Element of NAT st x = [i,j] holds
Im f2,x = [:(Im (Nbdl1 n),i),(Im (Nbdl1 m),j):] ) implies f1 = f2 )
assume that
A10: for x being set st x in [:(Seg n),(Seg m):] holds
for i, j being Element of NAT st x = [i,j] holds
Im f1,x = [:(Im (Nbdl1 n),i),(Im (Nbdl1 m),j):] and
A11: for x being set st x in [:(Seg n),(Seg m):] holds
for i, j being Element of NAT st x = [i,j] holds
Im f2,x = [:(Im (Nbdl1 n),i),(Im (Nbdl1 m),j):] ; :: thesis: f1 = f2
for x being set st x in [:(Seg n),(Seg m):] holds
Im f1,x = Im f2,x

proof

let x be set ; :: thesis: ( x in [:(Seg n),(Seg m):] implies Im f1,x = Im f2,x )
assume A12: x in [:(Seg n),(Seg m):] ; :: thesis: Im f1,x = Im f2,x
then consider u, y being set such that
A13: ( u in Seg n & y in Seg m ) and
A14: x = [u,y] by ZFMISC_1:def 2;
reconsider i = u, j = y as Element of NAT by A13;
Im f1,x = [:(Im (Nbdl1 n),i),(Im (Nbdl1 m),j):] by A10, A12, A14;
hence Im f1,x = Im f2,x by A11, A12, A14; :: thesis: verum

end;

hence f1 = f2 by RELSET_1:54; :: thesis: verum