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 Th43:
  for G1 being _Graph, G2 being Subgraph of G1, G3 being Subgraph
  of G2 holds G3 is Subgraph of G1
proof
  let G1 be _Graph, G2 be Subgraph of G1, G3 be Subgraph of G2;
A1: the_Edges_of G2 c= the_Edges_of G1;
A2: the_Vertices_of G3 c= the_Vertices_of G2;
  now
    thus the_Vertices_of G3 c= the_Vertices_of G1 by A2,XBOOLE_1:1;
    thus the_Edges_of G3 c= the_Edges_of G1 by A1;
    let e be set;
    assume
A3: e in the_Edges_of G3;
    hence (the_Source_of G3).e = (the_Source_of G2).e by Def32
      .= (the_Source_of G1).e by A3,Def32;
    thus (the_Target_of G3).e = (the_Target_of G2).e by A3,Def32
      .= (the_Target_of G1).e by A3,Def32;
  end;
  hence thesis by Def32;
end;
