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 Th37:
  for X being set, b1, b2 being natural-valued ManySortedSet of X
  holds support (b1+b2) = support b1 \/ support b2
proof
  let X be set, b1, b2 be natural-valued ManySortedSet of X;
  now
    let x be object;
    hereby
      assume x in support b1 \/ support b2;
      then x in support b1 or x in support b2 by XBOOLE_0:def 3;
      then b1.x <> 0 or b2.x <> 0 by Def7;
      then b1.x + b2.x <> 0;
      hence (b1+b2).x <> 0 by Def5;
    end;
    assume
A1: (b1+b2).x <> 0;
    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;
  hence thesis by Def7;
end;
