
theorem
  not ex p being Path of KoenigsbergBridges st p is non cyclic Eulerian
  proof
    given p being Path of KoenigsbergBridges such that
    p: p is non cyclic Eulerian;
    consider v1,v2 being Vertex of KoenigsbergBridges such that
    v: v1<>v2 & for v being Vertex of KoenigsbergBridges holds
    Degree v is even iff v<>v1 & v<>v2 by p,GRAPH_3:60;
    (v1=0 or v1=1 or v1=2 or v1=3) & (v2=0 or v2=1 or v2=2 or v2=3) & v1<>v2
    by v,ENUMSET1:def 2;
    then per cases;
    suppose s: v1=0 & v2=1 or v1=0 & v2=2 or v1=1 & v2=0 or
      v1=1 & v2=2 or v1=2 & v2=0 or v1=2 & v2=1;
      reconsider v=3 as Vertex of KoenigsbergBridges by ENUMSET1:def 2;
      Degree v = 3 by d3; then
      Degree v is not even by POLYFORM:6;
      hence contradiction by s,v;
    end;
    suppose s: v1=1 & v2=3 or v1=2 & v2=3 or v1=3 & v2=1 or v1=3 & v2=2;
      reconsider v=0 as Vertex of KoenigsbergBridges by ENUMSET1:def 2;
      Degree v = 3 by d0; then
      Degree v is not even by POLYFORM:6;
      hence contradiction by s,v;
    end;
    suppose s: v1=0 & v2=3 or v1=3 & v2=0;
      reconsider v=1 as Vertex of KoenigsbergBridges by ENUMSET1:def 2;
      Degree v = 3 by d1; then
      Degree v is not even by POLYFORM:6;
      hence contradiction by s,v;
    end;
  end;
