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 G being _Graph, V1,V2,E1,E2 being set, G1 being inducedSubgraph of
G,V1,E1, G2 being inducedSubgraph of G,V2,E2 st V2 c= V1 & E2 c= E1 & V2 is non
  empty Subset of the_Vertices_of G & E2 c= G.edgesBetween(V2) holds G2 is
  Subgraph of G1
proof
  let G be _Graph, V1,V2,E1,E2 be set, G1 be inducedSubgraph of G,V1,E1, G2 be
  inducedSubgraph of G,V2,E2;
  assume that
A1: V2 c= V1 & E2 c= E1 and
A2: V2 is non empty Subset of the_Vertices_of G & E2 c= G.edgesBetween( V2);
A3: the_Vertices_of G2 = V2 & the_Edges_of G2 = E2 by A2,Def37;
  now
    per cases;
    suppose
      V1 is non empty Subset of the_Vertices_of G & E1 c= G .edgesBetween(V1);
      then the_Vertices_of G1 = V1 & the_Edges_of G1 = E1 by Def37;
      hence thesis by A1,A3,Th44;
    end;
    suppose
      not (V1 is non empty Subset of the_Vertices_of G & E1 c= G
      .edgesBetween(V1));
      then G1 == G by Def37;
      hence thesis by A3,Th44;
    end;
  end;
  hence thesis;
end;
