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 Th51:
  for G1 being _Graph, e being set, G2 being removeEdge of G1,e
holds the_Vertices_of G2 = the_Vertices_of G1 & the_Edges_of G2 = the_Edges_of
  G1 \ {e}
proof
  let G1 be _Graph, e be set, G2 be removeEdge of G1,e;
  set V = the_Vertices_of G1;
  V c= V;
  then reconsider V as non empty Subset of the_Vertices_of G1;
  set E = the_Edges_of G1 \ {e};
  reconsider E as Subset of G1.edgesBetween(V) by Th34;
  G2 is inducedSubgraph of G1,V,E;
  hence thesis by Def37;
end;
