
theorem Th17:
  for G being _finite _Graph, S being VNumberingSeq of G, m, n
  being Nat st m <= n holds S.m c= S.n
proof
  let G be _finite _Graph, S be VNumberingSeq of G, m,n be Nat;
  assume m <= n;
  then
A1: ex j being Nat st n = m + j by NAT_1:10;
  set CSM = S.m;
  set VLM = CSM;
  defpred P[Nat] means VLM c= S.(m+$1);
A2: now
    let k be Nat;
    set CSK = S.k;
    set VLK = CSK;
    set CK1 = S.(k+1);
    set VK1 = CK1;
    per cases;
    suppose
A3:   k < S.Lifespan();
      set w = S.PickedAt(k);
      set wf = w .--> (S.Lifespan() -' k);
      not w in dom VLK by A3,Def9;
      then
A5:   dom wf misses dom VLK by ZFMISC_1:50;
      VK1 = VLK +* (w .--> (S.Lifespan()-'k)) by A3,Def9;
      hence VLK c= VK1 by A5,FUNCT_4:32;
    end;
    suppose
A6:   S.Lifespan() <= k;
      k <= k+1 by NAT_1:13;
      hence VLK c= VK1 by A6,Th10;
    end;
  end;
A7: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat such that
A8: P[k];
    S.(m+k) c= S.(m+k+1) by A2;
    hence thesis by A8,XBOOLE_1:1;
  end;
A9: P[ 0 ];
  for k being Nat holds P[k] from NAT_1:sch 2(A9,A7);
  hence thesis by A1;
end;
