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 _Graph, G2 being Subgraph of G1, v1 being Vertex of G1,
v2 being Vertex of G2 st v1 = v2 & v1 is endvertex holds v2 is endvertex or v2
  is isolated
proof
  let G1 be _Graph, G2 be Subgraph of G1, v1 be Vertex of G1, v2 be Vertex of
  G2;
  assume that
A1: v1 = v2 and
A2: v1 is endvertex;
  consider e being object such that
A3: v1.edgesInOut() = {e} and
A4: not e Joins v1,v1,G1 by A2;
  v2.edgesInOut() c= v1.edgesInOut() by A1,Th78;
  then
A5: v2.edgesInOut() = {} or v2.edgesInOut() = {e} by A3,ZFMISC_1:33;
    v2 is not isolated implies v2 is endvertex by A5,A1,A4,Lm4;
  hence thesis;
end;
