
theorem Th67:
  for G1 being _Graph, G2 being removeLoops of G1
  for v1 being Vertex of G1, v2 being Vertex of G2
  st v1 = v2 holds v1 is cut-vertex iff v2 is cut-vertex
proof
  let G1 be _Graph, G2 be removeLoops of G1;
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A1: v1 = v2;
  hereby
    assume A2: v1 is cut-vertex;
    now
      let G4 be removeVertex of G2, v2;
      set G3 = the removeVertex of G1, v1;
      G4 is removeLoops of G3 by A1, Th66;
      then A3: G3.numComponents() = G4.numComponents() by Th64;
      G1.numComponents() in G3.numComponents() by A2, GLIB_002:def 10;
      hence G2.numComponents() in G4.numComponents() by A3, Th64;
    end;
    hence v2 is cut-vertex by GLIB_002:def 10;
  end;
  assume A4: v2 is cut-vertex;
  now
    let G3 be removeVertex of G1, v1;
    set G4 = the removeVertex of G2, v2;
    G4 is removeLoops of G3 by A1, Th66;
    then A5: G3.numComponents() = G4.numComponents() by Th64;
    G2.numComponents() in G4.numComponents() by A4, GLIB_002:def 10;
    hence G1.numComponents() in G3.numComponents() by A5, Th64;
  end;
  hence v1 is cut-vertex by GLIB_002:def 10;
end;
