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, b1, b2 being real-valued ManySortedSet of X holds
  support (b1+b2) c= support b1 \/ support b2
proof
  let X be set, b1, b2 be real-valued ManySortedSet of X;
  let x be object;
  assume x in support (b1+b2);
  then
A1: (b1+b2).x <> 0 by Def7;
  assume
A2: not x in support b1 \/ support b2;
  then not x in support b1 by XBOOLE_0:def 3;
  then
A3: b1.x = 0 by Def7;
  not x in support b2 by A2,XBOOLE_0:def 3;
  then b1.x+b2.x = 0 by A3,Def7;
  hence contradiction by A1,Def5;
end;
