reserve A for set, x,y,z for object,
  k for Element of NAT;
reserve n for Nat,
  x for object;
reserve V, C for set;

theorem
  for X being set, b, b1, b2 being real-valued ManySortedSet of X
   st for x being object st x in X holds b.x = b1.x+b2.x
  holds b = b1+b2
proof
  let X be set, b, b1, b2 be real-valued ManySortedSet of X;
  assume
A1: for x being object st x in X holds b.x = b1.x+b2.x;
  now
    let x be object;
    per cases;
    suppose
      x in X;
      hence b.x = b1.x+b2.x by A1;
    end;
    suppose
A2:   not x in X;
A3:   dom b2 = X by PARTFUN1:def 2;
A4:   dom b1 = X by PARTFUN1:def 2;
      dom b = X by PARTFUN1:def 2;
      hence b.x = 0 qua Nat + (0 qua Nat) by A2,FUNCT_1:def 2
        .= 0 qua Nat+b2.x by A2,A3,FUNCT_1:def 2
        .= b1.x+b2.x by A2,A4,FUNCT_1:def 2;
    end;
  end;
  hence thesis by Def5;
end;
