
theorem Th20:
  for G being _finite _Graph, S being VNumberingSeq of G, m,n being
Nat, v being set st v in dom (S.m) & (S.m).v = n holds S.PickedAt(S.Lifespan()
  -'n) = v
proof
  let G be _finite _Graph, S be VNumberingSeq of G, m,n be Nat, v be set;
  set CSM = S.m;
  set VLM = CSM;
  set j = S.Lifespan() -' n;
  set CJ1 = S.(j+1);
  set VJ1 = CJ1;
  assume that
A1: v in dom CSM and
A2: VLM.v = n;
  set w = S.PickedAt(j);
  n <= S.Lifespan() by A2,Th15;
  then
A3: S.Lifespan() -' n = S.Lifespan() - n by XREAL_1:233;
A4: 0 < n by A1,A2,Th15;
  then
A5: j < S.Lifespan() by A3,XREAL_1:44;
  then S.Lifespan() -' j = S.Lifespan() - (S.Lifespan() - n) by A3,XREAL_1:233;
  then
A6: VJ1.w = n by A4,A3,Th12,XREAL_1:44;
A7: VLM is one-to-one by Th18;
A8: w in dom CJ1 by A5,Th11;
  per cases;
  suppose
A9: m <= j;
    then m + n <= S.Lifespan() - n + n by A3,XREAL_1:6;
    then
A10: n + m - m <= S.Lifespan() - m by XREAL_1:9;
A11: rng VLM = (Seg S.Lifespan()) \ Seg (S.Lifespan() -' m) by Th14;
A12: n in rng VLM by A1,A2,FUNCT_1:def 3;
    S.Lifespan() - m = S.Lifespan() -' m by A5,A9,XREAL_1:233,XXREAL_0:2;
    hence thesis by A10,A12,A11,Th3;
  end;
  suppose
    j < m;
    then j+1 <= m by NAT_1:13;
    then
A13: VJ1 c= VLM by Th17;
    then
A14: dom VJ1 c= dom VLM by RELAT_1:11;
    [w,n] in VJ1 by A8,A6,FUNCT_1:def 2;
    then VLM.w = n by A8,A13,A14,FUNCT_1:def 2;
    hence thesis by A1,A2,A8,A7,A14,FUNCT_1:def 4;
  end;
end;
