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 Th93:
  for G1,G2 being inducedSubgraph of G,V,E holds G1 == G2
proof
  let G1,G2 be inducedSubgraph of G,V,E;
  now
    per cases;
    suppose
A1:   V is non empty Subset of the_Vertices_of G & E c= G.edgesBetween (V);
      then
A2:   the_Edges_of G1 = E & the_Edges_of G2 = E by Def37;
      the_Vertices_of G1 = V & the_Vertices_of G2 = V by A1,Def37;
      hence thesis by A2,Th86;
    end;
    suppose
      not (V is non empty Subset of the_Vertices_of G & E c= G
      .edgesBetween(V));
      then G1 == G & G2 == G by Def37;
      hence thesis;
    end;
  end;
  hence thesis;
end;
