
theorem Th44:
  for G being _finite _Graph, x being set, i,j being Nat st i <= j
  holds ((LexBFS:CSeq(G)).i)`2.x c= ((LexBFS:CSeq(G)).j)`2.x
proof
  let G be _finite _Graph, x be set, i,j be Nat;
  assume i <= j;
  then
A1: ex k being Nat st j = i+k by NAT_1:10;
  set CS = (LexBFS:CSeq(G));
  set CSI = CS.i, V2I = CSI`2;
  defpred P[Nat] means V2I.x c= (CS.(i+$1))`2.x;
A2: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat such that
A3: P[k];
    set CK1 = CS.(i+k+1);
    set CSK = CS.(i+k);
    set V2K = CSK`2;
    set VLK = CSK`1;
    set V21 = CK1`2;
    per cases;
    suppose
      i+k+1 <= G.order();
      then i+k < G.order() by NAT_1:13;
      then consider w being Vertex of G such that
      w = LexBFS:PickUnnumbered(CSK) and
A4:   for v being set holds (v in G.AdjacentSet({w}) & not v in dom
VLK implies V21.v = V2K.v \/ {G.order() -' (i+k)}) & (not v in G.AdjacentSet({w
      }) or v in dom VLK implies V21.v = V2K.v) by Th42;
      per cases;
      suppose
        x in G.AdjacentSet({w}) & not x in dom VLK;
        then V21.x = V2K.x \/ {G.order() -' (i+k)} by A4;
        then V2K.x c= V21.x by XBOOLE_1:7;
        hence thesis by A3,XBOOLE_1:1;
      end;
      suppose
        not x in G.AdjacentSet({w}) or x in dom VLK;
        hence thesis by A3,A4;
      end;
    end;
    suppose
A5:   G.order() < i+k+1;
A6:   i+k <= i+k+1 by NAT_1:13;
      G.order() <= i+k by A5,NAT_1:13;
      hence thesis by A3,A6,Th34;
    end;
  end;
A7: P[ 0 ];
  for k being Nat holds P[k] from NAT_1:sch 2(A7,A2);
  hence thesis by A1;
end;
