
theorem Th42:
  for G being _finite connected real-weighted WGraph, n being Nat
  holds (PRIM:CompSeq(G).n)`2 c= (PRIM:MST(G))`2
proof
  let G be _finite connected real-weighted WGraph, n be Nat;
  set PCS = PRIM:CompSeq(G);
  defpred P[Nat] means (PCS.(PCS.Lifespan()+$1)) = PRIM:MST(G);
A1: now
    set off = PCS.Lifespan();
    let n be Nat;
    set Gn = PCS.(off+n), Gn1 = PCS.(off+n+1);
    set Next = PRIM:NextBestEdges(Gn), e = the Element of Next;
A2: Gn1 = PRIM:Step(Gn) by Def17;
    assume
A3: P[n];
    then
A4: Gn`1 = the_Vertices_of G by Th39;
    now
      assume Next <> {};
      then
      ex v being Vertex of G st ( not v in Gn`1)& Gn1 = [ Gn`1 \/ {v}, Gn`2
      \/ {e} ] by A2,Th28;
      hence contradiction by A4;
    end;
    hence P[n+1] by A3,A2,Def15;
  end;
A5: P[ 0 ];
A6: for n being Nat holds P[n] from NAT_1:sch 2(A5,A1);
  now
    per cases;
    suppose
      n <= PCS.Lifespan();
      hence thesis by Th34;
    end;
    suppose
      PCS.Lifespan() < n;
      then ex k being Nat st n = PCS.Lifespan() + k by NAT_1:10;
      hence thesis by A6;
    end;
  end;
  hence thesis;
end;
