
theorem Th95: :: Chordal02
  for G being _finite _Graph st card the_Vertices_of G <= 3 holds G is chordal
proof
  let G be _finite _Graph such that
A1: card the_Vertices_of G <= 3;
  now
    reconsider n4=2*3+1 as odd Nat;
    reconsider n3=2*2+1 as odd Nat;
    reconsider n2=2*1+1 as odd Nat;
    reconsider n1=2*0+1 as odd Nat;
    let W be Walk of G such that
A2: W.length() > 3 and
A3: W is Cycle-like;
    set x3=W.n3;
    set x2=W.n2;
    set x4=W.n4;
    set x1=W.n1;
    W.length() >= 3+1 by A2,NAT_1:13;
    then 2*W.length() >= 2*4 by XREAL_1:64;
    then 2*W.length() + 1 >= 8 + 1 by XREAL_1:7;
    then
A4: len W >= 9 by GLIB_001:112;
    then
A5: n4 < len W by XXREAL_0:2;
    then
A6: x1 <> x4 by A3,GLIB_001:def 28;
    n2 < len W by A4,XXREAL_0:2;
    then
A7: x1 <> x2 by A3,GLIB_001:def 28;
A8: n3 < len W by A4,XXREAL_0:2;
    then
A9: x2 <> x3 by A3,GLIB_001:def 28;
A10: x3 <> x4 by A3,A5,GLIB_001:def 28;
A11: x2 <> x4 by A3,A5,GLIB_001:def 28;
    now
      let x be object;
      assume
A12:  x in {x1,x2,x3,x4};
      now
        per cases by A12,ENUMSET1:def 2;
        suppose
          x=x1;
          hence x in the_Vertices_of G by A4,GLIB_001:7,XXREAL_0:2;
        end;
        suppose
          x=x2;
          hence x in the_Vertices_of G by A4,GLIB_001:7,XXREAL_0:2;
        end;
        suppose
          x=x3;
          hence x in the_Vertices_of G by A4,GLIB_001:7,XXREAL_0:2;
        end;
        suppose
          x=x4;
          hence x in the_Vertices_of G by A4,GLIB_001:7,XXREAL_0:2;
        end;
      end;
      hence x in the_Vertices_of G;
    end;
    then
A13: {x1,x2,x3,x4} c= the_Vertices_of G;
    x1 <> x3 by A3,A8,GLIB_001:def 28;
    then card {x1,x2,x3,x4} = 4 by A7,A6,A9,A11,A10,CARD_2:59;
    then 4 <= card the_Vertices_of G by A13,NAT_1:43;
    hence contradiction by A1,XXREAL_0:2;
  end;
  then for W being Walk of G st W.length() > 3 & W is Cycle-like holds W is
  chordal;
  hence thesis;
end;
