
theorem d3:
  for v being Vertex of KoenigsbergBridges st v=3 holds Degree v = 3
  proof
    let v be Vertex of KoenigsbergBridges such that
    v: v=3;
    c1: Edges_In(v)={30,60,70}
    proof
      thus Edges_In(v)c={30,60,70}
      proof
        let a be object;
        reconsider s=a as set by TARSKI:1;
        assume a in Edges_In(v);
        then s: s in KEdges & KTarget.s=v by GRAPH_3:def 1;
        dom KTarget = KEdges by FUNCT_2:def 1; then
        s in dom KTarget by s; then
        [s,v] in KTarget by s,FUNCT_1:1; then
        [s,v]=[10,1] or [s,v]=[20,2] or [s,v]=[30,3] or [s,v]=[40,2] or
        [s,v]=[50,2] or [s,v]=[60,3] or [s,v]=[70,3] by ENUMSET1:def 5;
        then s=30 or s=60 or s=70 by v,XTUPLE_0:1;
        hence a in {30,60,70} by ENUMSET1:def 1;
      end;
      let a be object;
      reconsider s=a as set by TARSKI:1;
      assume a in {30,60,70}; then
      a: a=30 or a=60 or a=70 by ENUMSET1:def 1; then
      s: s in KEdges by ENUMSET1:def 5;
      [s,v] in KTarget by v,a,ENUMSET1:def 5; then
      KTarget.s=v by FUNCT_1:1;
      hence a in Edges_In(v) by s,GRAPH_3:def 1;
    end;
    c2: Edges_Out(v)={}
    proof
      thus Edges_Out(v)c={}
      proof
        let a be object;
        reconsider s=a as set by TARSKI:1;
        assume a in Edges_Out(v);
        then s: s in KEdges & KSource.s=v by GRAPH_3:def 2;
        dom KSource = KEdges by FUNCT_2:def 1; then
        s in dom KSource by s; then
        [s,v] in KSource by s,FUNCT_1:1; then
        [s,v]=[10,0] or [s,v]=[20,0] or [s,v]=[30,0] or [s,v]=[40,1] or
        [s,v]=[50,1] or [s,v]=[60,2] or [s,v]=[70,2] by ENUMSET1:def 5;
        hence a in {} by v,XTUPLE_0:1;
      end;
      let a be object;
      assume a in {};
      hence a in Edges_Out(v);
    end;
    card Edges_In(v)=3 & card Edges_Out(v)=0 by c1,c2,CARD_2:58;
    then Degree(v, the carrier' of KoenigsbergBridges) = 3+0;
    hence Degree v = 3 by GRAPH_3:24;
  end;
