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 Th47:
  for G1 being non _trivial _Graph, v being Vertex of G1, G2 being
  removeVertex of G1,v holds the_Vertices_of G2 = the_Vertices_of G1 \ {v} &
  the_Edges_of G2 = G1.edgesBetween(the_Vertices_of G1 \ {v})
proof
  let G1 be non _trivial _Graph,v be Vertex of G1,
      G2 be removeVertex of G1,v;
  set VG = the_Vertices_of G1, V = VG \ {v};
  now
    assume V is empty;
    then VG c= {v} by XBOOLE_1:37;
    then VG = {v} by ZFMISC_1:33;
    then card VG = 1 by CARD_1:30;
    hence contradiction by Def19;
  end;
  then reconsider V as non empty Subset of VG;
  G2 is inducedSubgraph of G1,V;
  hence thesis by Def37;
end;
