reserve X,Z for set;
reserve x,y,z for object;
reserve A,B,C for Subset of X;

theorem Th61:
  for A being Order, f being finite-support Function of A, REAL holds
    (eqSumOf f)*(proj A) = f
proof
  let A be Order, f be finite-support Function of A, REAL;
  set F = (eqSumOf f)*(proj A);
  QuotientOrder(A) is empty implies A is empty;
  then dom F = the carrier of A by FUNCT_2:def 1;
  then A1: dom f = dom F by FUNCT_2:def 1;
  for x being object st x in dom f holds f.x = F.x
  proof
    let x be object;
    assume x in dom f;
    then reconsider z = x as Element of A;
    for y being Element of A st z =~ y holds z = y by ORDERS_2:2;
    hence thesis by Th60;
  end;
  hence thesis by A1, FUNCT_1:2;
end;
