reserve G,G1,G2 for _Graph;
reserve e,x,y for set;
reserve v,v1,v2 for Vertex of G;
reserve W for Walk of G;

theorem
  for v1 being Vertex of G1, v2 being Vertex of G2 st G1 == G2 & v1 = v2
  holds v1 is cut-vertex implies v2 is cut-vertex
proof
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume that
A1: G1 == G2 and
A2: v1 = v2 and
A3: v1 is cut-vertex;
A4: G1 is non _trivial by A3,Th37;
  then
A5: G2 is non _trivial by A1,GLIB_000:89;
  let G2A be removeVertex of G2,v2;
  set G1A = the removeVertex of G1,v1;
  G1.numComponents() = G2.numComponents() by A1,Lm12;
  then
A6: G2.numComponents() in G1A.numComponents() by A3;
  the_Vertices_of G1A = the_Vertices_of G1 \ {v2} by A2,A4,GLIB_000:47
    .= the_Vertices_of G2 \ {v2} by A1,GLIB_000:def 34;
  then
A7: the_Vertices_of G2A = the_Vertices_of G1A by A5,GLIB_000:47;
  G2 is Subgraph of G1 by A1,GLIB_000:87;
  then
A8: G2A is Subgraph of G1 by GLIB_000:43;
  the_Edges_of G1A = G1.edgesBetween(the_Vertices_of G1 \ {v1}) by A4,
GLIB_000:47
    .= G1.edgesBetween(the_Vertices_of G2 \ {v2}) by A1,A2,GLIB_000:def 34
    .= G2.edgesBetween(the_Vertices_of G2 \ {v2}) by A1,GLIB_000:90;
  then the_Edges_of G2A = the_Edges_of G1A by A5,GLIB_000:47;
  hence thesis by A6,A7,A8,Lm12,GLIB_000:86;
end;
