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 G being _trivial _Graph, H being _Graph
  st the_Vertices_of H c= the_Vertices_of G & the_Edges_of H c= the_Edges_of G
  holds H is _trivial & H is Subgraph of G
proof
  let G be _trivial _Graph, H be _Graph;
  assume A1: the_Vertices_of H c= the_Vertices_of G &
    the_Edges_of H c= the_Edges_of G;
  consider v being Vertex of G such that
    A2: the_Vertices_of G = {v} by Th22;
  A3: the_Vertices_of H = {v} by A1, A2, ZFMISC_1:33;
  then card the_Vertices_of H = 1 by CARD_1:30;
  hence H is _trivial;
  now
    let e be set;
    assume A4: e in the_Edges_of H;
    then (the_Source_of H).e in the_Vertices_of H &
      (the_Target_of H).e in the_Vertices_of H by FUNCT_2:5;
    then A5: (the_Source_of H).e = v & (the_Target_of H).e = v
      by A3, TARSKI:def 1;
    (the_Source_of G).e in the_Vertices_of G &
      (the_Target_of G).e in the_Vertices_of G by A1, A4, FUNCT_2:5;
    hence (the_Source_of H).e = (the_Source_of G).e &
      (the_Target_of H).e = (the_Target_of G).e by A2, A5, TARSKI:def 1;
  end;
  hence thesis by A1, Def32;
end;
