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 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 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 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 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)

for i, j being 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 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 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 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

hence
f1 = f2
by RELSET_1:31; :: thesis: verum
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 object 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 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;assume A12: x in [:(Seg n),(Seg m):] ; :: thesis: Im (f1,x) = Im (f2,x)

then consider u, y being object 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 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