
theorem Th58: :: AdjGraph00
  for G1, G2 be _Graph st G1 == G2 for u1 being Vertex of G1, u2
  being Vertex of G2 st u1 = u2 for H1 being AdjGraph of G1,{u1}, H2 being
  AdjGraph of G2,{u2} holds H1 == H2
proof
  let G1,G2 be _Graph such that
A1: G1 == G2;
  let u1 be Vertex of G1, u2 be Vertex of G2 such that
A2: u1 = u2;
  set G2Adj = G2.AdjacentSet({u2});
  set G1Adj = G1.AdjacentSet({u1});
A3: G1Adj = G2Adj by A1,A2,Th50;
  let H1 be AdjGraph of G1,{u1};
  let H2 be AdjGraph of G2,{u2};
A4: H1 is inducedSubgraph of G1,G1Adj by Def5;
A5: H2 is inducedSubgraph of G2,G2Adj by Def5;
  per cases;
  suppose
A6: not G1Adj is non empty Subset of the_Vertices_of G1;
    then H1 == G1 by A4,GLIB_000:def 37;
    then
A7: H1 == G2 by A1;
    H2 == G2 by A5,A3,A6,GLIB_000:def 37;
    hence thesis by A7;
  end;
  suppose
A8: G1Adj is non empty Subset of the_Vertices_of G1;
    then the_Vertices_of H1 = G1Adj by A4,GLIB_000:def 37;
    then
A9: the_Vertices_of H1 = the_Vertices_of H2 by A5,A3,GLIB_000:def 37;
    G1 is Subgraph of G2 by A1,GLIB_000:87;
    then
A10: G1.edgesBetween(G1Adj) c= G2.edgesBetween(G1Adj) by GLIB_000:76;
A11: the_Edges_of H1 = G1.edgesBetween(G1Adj) by A4,A8,GLIB_000:def 37;
    G2 is Subgraph of G1 by A1,GLIB_000:87;
    then
A12: G2.edgesBetween(G1Adj) c= G1.edgesBetween(G1Adj) by GLIB_000:76;
    G2 is Subgraph of G1 by A1,GLIB_000:87;
    then
A13: H2 is Subgraph of G1 by GLIB_000:43;
    the_Edges_of H2 = G2.edgesBetween(G2Adj) by A5,A3,A8,GLIB_000:def 37;
    then the_Edges_of H1 = the_Edges_of H2 by A3,A11,A10,A12;
    hence thesis by A9,A13,GLIB_000:86;
  end;
end;
