
theorem Th29:
  for G being real-weighted WGraph, L be PRIM:Labeling of G holds
  L`1 c= (PRIM:Step(L))`1 & L`2 c= (PRIM:Step(L))`2
proof
  let G be real-weighted WGraph, L be PRIM:Labeling of G;
  set G2 = PRIM:Step(L);
  set Next = PRIM:NextBestEdges(L), e = the Element of Next;
  now
    per cases;
    suppose
      Next = {};
      hence thesis by Def15;
    end;
    suppose
      Next <> {};
      then consider v being Vertex of G such that
      not v in L`1 and
A1:   G2 = [L`1 \/ {v}, L`2 \/ {e}] by Th28;
      G2`1 = L`1 \/ {v} by A1;
      hence L`1 c= G2`1 by XBOOLE_1:7;
      G2`2 = L`2 \/ {e} by A1;
      hence L`2 c= G2`2 by XBOOLE_1:7;
    end;
  end;
  hence thesis;
end;
