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 G1 being _Graph, G2 being inducedSubgraph of G1,the_Vertices_of G1
  holds G1 == G2
proof
  let G1 be _Graph, G2 be inducedSubgraph of G1, the_Vertices_of G1;
A1: the_Vertices_of G1 c= the_Vertices_of G1;
  then the_Edges_of G2 = G1.edgesBetween(the_Vertices_of G1) by Def37;
  then
A2: the_Edges_of G2 = the_Edges_of G1 by Th34;
  then the_Source_of G2 = (the_Source_of G1)|the_Edges_of G1 by Th45;
  then the_Source_of G2 = (the_Source_of G1)|(dom the_Source_of G1);
  then
A3: the_Source_of G2 = the_Source_of G1;
  the_Target_of G2 = (the_Target_of G1)|the_Edges_of G1 by A2,Th45;
  then the_Target_of G2 = (the_Target_of G1)|(dom the_Target_of G1);
  then
A4: the_Target_of G2 = the_Target_of G1;
  the_Vertices_of G2 = the_Vertices_of G1 by A1,Def37;
  hence thesis by A2,A3,A4;
end;
