
theorem Th52:
  for G being _finite _Graph, i,j being Nat,v being Vertex of G st
j in ((LexBFS:CSeq(G)).i)`2.v ex w being Vertex of G st w in dom ((LexBFS:CSeq(
  G)).i)`1 & (((LexBFS:CSeq(G)).i)`1).w = j & v in G.AdjacentSet{w}
proof
  let G be _finite _Graph, i,j be Nat;
  let v be Vertex of G;
  set CSI = (LexBFS:CSeq(G)).i;
  set VLI = (LexBFS:CSeq(G))``1.i;
  set V2I = CSI`2;
  set n = G.order() -' j;
  set CSN = (LexBFS:CSeq(G)).n;
  set VLN = CSN`1;
A1: G.order() = (LexBFS:CSeq(G)).Lifespan() by Th37;
A2: (LexBFS:CSeq(G)).Lifespan() = (LexBFS:CSeq(G)``1).Lifespan() by Th39;
  assume
A3: j in V2I.v;
  then consider w being Vertex of G such that
A4: LexBFS:PickUnnumbered(CSN) = w and
  not w in dom VLN and
A5: v in G.AdjacentSet({w}) by Th48;
A6: V2I.v c= Seg G.order() \ Seg (G.order() -' i) by Th43;
  then
A7: G.order() -' i < j by A3,Th3;
A8: j <= G.order() by A3,A6,Th3;
  then
A9: G.order() -' j = G.order() - j by XREAL_1:233;
  then
A10: n < G.order() by A7,XREAL_1:44;
A11: G.order() - n = G.order() - (G.order() - j) by A8,XREAL_1:233;
  then G.order() - i < G.order() - n by A7,XREAL_0:def 2;
  then G.order() - i + i < G.order() - n + i by XREAL_1:6;
  then G.order() + n < G.order() + i - n + n by XREAL_1:6;
  then
A12: n + G.order() - G.order() < i + G.order() - G.order() by XREAL_1:9;
A13: w = ((LexBFS:CSeq(G))``1).PickedAt(n) by A4,A7,A9,Th41,XREAL_1:44;
  then
A14: VLI.w = G.order()-'n by A10,A1,A2,A12,Th21;
A15: CSI`1 = VLI by Def15;
  then w in dom CSI`1 by A10,A1,A2,A13,A12,Th21;
  hence thesis by A15,A5,A10,A11,A14,XREAL_1:233;
end;
