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 Th49:
  for G1 being _Graph, V being set, G2 being removeVertices of G1,
V st V c< the_Vertices_of G1 holds the_Vertices_of G2 = the_Vertices_of G1 \ V
  & the_Edges_of G2 = G1.edgesBetween(the_Vertices_of G1 \ V)
proof
  let G1 be _Graph, V be set, G2 be removeVertices of G1,V;
  set VG2 = the_Vertices_of G1 \ V;
  assume
A1: V c< the_Vertices_of G1;
  now
    assume VG2 is empty;
    then the_Vertices_of G1 c= V by XBOOLE_1:37;
    hence contradiction by A1,XBOOLE_0:def 8;
  end;
  then reconsider VG2 as non empty Subset of the_Vertices_of G1;
  G2 is inducedSubgraph of G1,VG2;
  hence thesis by Def37;
end;
