let A, B be set ; :: thesis: for N, M being ManySortedSet of [:A,B:] st ( for i, j being object st i in A & j in B holds
N . (i,j) = M . (i,j) ) holds
M = N
let N, M be ManySortedSet of [:A,B:]; :: thesis: ( ( for i, j being object st i in A & j in B holds
N . (i,j) = M . (i,j) ) implies M = N )
assume A1: for i, j being object st i in A & j in B holds
N . (i,j) = M . (i,j) ; :: thesis: M = N

A2: now :: thesis: for x being object st x in [:A,B:] holds
N . x = M . x

let x be object ; :: thesis: ( x in [:A,B:] implies N . x = M . x )
assume A3: x in [:A,B:] ; :: thesis: N . x = M . x
then reconsider A1 = A, B1 = B as non empty set ;
consider i being Element of A1, j being Element of B1 such that
A4: x = [i,j] by A3, DOMAIN_1:1;
thus N . x = N . (i,j) by A4
.= M . (i,j) by A1
.= M . x by A4 ; :: thesis: verum

end;

( dom M = [:A,B:] & dom N = [:A,B:] ) by PARTFUN1:def 2;
hence M = N by A2, FUNCT_1:2; :: thesis: verum