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 G1 being non _trivial _Graph, v being Vertex of G1
  for G2 being removeVertex of G1, v
  for v1 being Vertex of G1, v2 being Vertex of G2 st v1 = v2 holds
    v2.edgesIn() = v1.edgesIn() \ v.edgesOut() &
    v2.edgesOut() = v1.edgesOut() \ v.edgesIn() &
    v2.edgesInOut() = v1.edgesInOut() \ v.edgesInOut()
proof
  let G1 be non _trivial _Graph, v be Vertex of G1;
  let G2 be removeVertex of G1, v;
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A1: v1 = v2;
  (the_Vertices_of G1) \ {v} <> {} by Th20;
  then the_Vertices_of G1 <> {v} by XBOOLE_1:37;
  then {v} c< the_Vertices_of G1 by XBOOLE_0:def 8;
  hence thesis by A1, Th166;
end;
