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 Th44:
  for G being _Graph, G1, G2 being Subgraph of G st
  the_Vertices_of G1 c= the_Vertices_of G2 & the_Edges_of G1 c= the_Edges_of G2
  holds G1 is Subgraph of G2
proof
  let G be _Graph, G1, G2 be Subgraph of G;
  assume that
A1: the_Vertices_of G1 c= the_Vertices_of G2 and
A2: the_Edges_of G1 c= the_Edges_of G2;
  now
    let e be set;
    assume
A3: e in the_Edges_of G1;
    hence (the_Source_of G1).e = (the_Source_of G).e by Def32
      .= (the_Source_of G2).e by A2,A3,Def32;
    thus (the_Target_of G1).e = (the_Target_of G).e by A3,Def32
      .= (the_Target_of G2).e by A2,A3,Def32;
  end;
  hence thesis by A1,A2,Def32;
end;
