reserve i,j,k,x for object;

theorem Th2:
  for A,B being set, N,M being ManySortedSet of [:A,B:] st for i,j
  st i in A & j in B holds N.(i,j) = M.(i,j) holds M = N
proof
  let A,B be set, N,M be ManySortedSet of [:A,B:];
  assume
A1: for i,j st i in A & j in B holds N.(i,j) = M.(i,j);
A2: now
    let x be object;
    assume
A3: x in [:A,B:];
    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;
  end;
  dom M = [:A,B:] & dom N = [:A,B:] by PARTFUN1:def 2;
  hence thesis by A2,FUNCT_1:2;
end;
