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 Th101:
  for G1,G3 being _Graph, V,E being set, G2 being inducedSubgraph of G1,V,E
  st G2 == G3 holds G3 is inducedSubgraph of G1,V,E
proof
  let G1,G3 be _Graph, V,E be set;
  let G2 be inducedSubgraph of G1,V,E;
  assume A1: G2 == G3;
  then G3 is Subgraph of G2 by Th87;
  then A2: G3 is Subgraph of G1 by Th43;
  per cases;
  suppose A3: V is non empty Subset of the_Vertices_of G1 &
    E c= G1.edgesBetween(V);
    then the_Vertices_of G2 = V & the_Edges_of G2 = E by Def37;
    then the_Vertices_of G3 = V & the_Edges_of G3 = E by A1;
    hence thesis by A2, A3, Def37;
  end;
  suppose A4: not (V is non empty Subset of the_Vertices_of G1 &
    E c= G1.edgesBetween(V));
    then G2 == G1 by Def37;
    then G3 == G1 by A1;
    hence thesis by A2, A4, Def37;
  end;
end;
