
theorem Th75:
  for G being _finite _Graph, n being Nat st n < G.order() ex w
being Vertex of G st w = MCS:PickUnnumbered((MCS:CSeq(G)).n) & for v being set
holds (v in G.AdjacentSet({w}) & not v in dom (((MCS:CSeq(G)).n)`1) implies (((
  MCS:CSeq(G)).(n+1))`2).v = (((MCS:CSeq(G)).n)`2).v + 1) & (not v in G
.AdjacentSet({w}) or v in dom (((MCS:CSeq(G)).n)`1) implies (((MCS:CSeq(G)).(n+
  1))`2).v = (((MCS:CSeq(G)).n)`2).v)
proof
  let G be _finite _Graph, n be Nat such that
A1: n < G.order();
  set CSN = (MCS:CSeq(G)).n;
  set VLN = CSN`1;
A2: n = card dom VLN by A1,Th65;
  set k = G.order() -' n;
  set w = MCS:PickUnnumbered(CSN);
  set CN1 = (MCS:CSeq(G)).(n+1);
  set CSlv = [ CSN`1 +* (w .--> k), CSN`2 ];
  set CSlv1 = CSN`1 +* (w .--> k);
A3: dom CSlv1 = dom (CSN`1) \/ {w} by Lm1;
  rng CSlv1 c= NAT;
  then CSlv1 in PFuncs(the_Vertices_of G, NAT) by A3,PARTFUN1:def 3;
  then reconsider CSlv as MCS:Labeling of G by ZFMISC_1:def 2;
A4: CN1 = MCS:Step(CSN) by Def25
    .= MCS:Update(CSN, w, n) by A1,A2,Def22
    .= MCS:LabelAdjacent(CSlv,w) by Def21;
  take w;
  set V2v = CSlv`2;
  set VLv = CSlv`1;
  set V21 = CN1`2;
  set V2N = CSN`2;
A5: dom VLv = (dom CSN`1) \/ {w} by Lm1;
  then
A6: dom VLN c= dom VLv by XBOOLE_1:7;
A7: now
    let v be set;
    assume
A8: not v in G.AdjacentSet({w}) or v in dom VLN;
    per cases by A8;
    suppose
      not v in G.AdjacentSet({w});
      hence V21.v = V2N.v by A4,Th60;
    end;
    suppose
      v in dom VLN;
      hence V21.v = V2N.v by A4,A6,Th61;
    end;
  end;
A9: dom V2N = the_Vertices_of G by FUNCT_2:def 1;
  now
    let v be set;
    assume that
A10: v in G.AdjacentSet({w}) and
A11: not v in dom VLN;
    not v in {w} by A10,CHORD:49;
    then not v in dom VLv by A5,A11,XBOOLE_0:def 3;
    hence V21.v = V2N.v + 1 by A4,A9,A10,Th62;
  end;
  hence thesis by A7;
end;
