reserve i,j,k,x for object;

theorem Th4:
  for A being set, N,M being ManySortedSet of [:A,A,A:] st for i,j
  ,k st i in A & j in A & k in A holds N.(i,j,k) = M.(i,j,k) holds M = N
proof
  let A be set, N,M be ManySortedSet of [:A,A,A:];
  assume
A1: for i,j,k st i in A & j in A & k in A holds N.(i,j,k) = M.(i,j,k);
A2: now
    let x be object;
    assume
A3: x in [:A,A,A:];
    then reconsider A as non empty set by MCART_1:31;
    consider i,j,k being Element of A such that
A4: x = [i,j,k] by A3,DOMAIN_1:3;
    thus M.x = M.(i,j,k) by A4,MULTOP_1:def 1
      .= N.(i,j,k) by A1
      .= N.x by A4,MULTOP_1:def 1;
  end;
  dom M = [:A,A,A:] & dom N = [:A,A,A:] by PARTFUN1:def 2;
  hence thesis by A2,FUNCT_1:2;
end;
