reserve GS for GraphStruct;
reserve G,G1,G2,G3 for _Graph;
reserve e,x,x1,x2,y,y1,y2,E,V,X,Y for set;
reserve n,n1,n2 for Nat;
reserve v,v1,v2 for Vertex of G;

theorem
  for G being _Graph, v being Vertex of G
  holds v is isolated iff v.allNeighbors() = {}
proof
  let G be _Graph, v be Vertex of G;
  hereby
    assume v is isolated;
    then v.edgesInOut() = {};
    then v.edgesIn() \/ v.edgesOut() = {};
    then v.edgesIn() = {} & v.edgesOut() = {};
    then (the_Source_of G).:v.edgesIn() = {} &
      (the_Target_of G).:v.edgesOut() = {};
    then v.inNeighbors() = {} & v.outNeighbors() = {};
    then v.inNeighbors() \/ v.outNeighbors() = {} \/ {};
    hence v.allNeighbors() = {};
  end;
  assume A1: v.allNeighbors() = {};
  assume not v is isolated;
  then A2: v.edgesInOut() <> {};
  set e = the Element of v.edgesInOut();
  consider v0 being Vertex of G such that
    A3: e Joins v,v0,G by A2, Th64;
  thus contradiction by A1, A3, Th71;
end;
