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 G1 being _Graph, G2 being Subgraph of G1, x,y,e being set st e in
the_Edges_of G2 holds (e Joins x,y,G1 implies e Joins x,y,G2) & (e DJoins x,y,
  G1 implies e DJoins x,y,G2) & (e SJoins x,y,G1 implies e SJoins x,y,G2) & (e
  DSJoins x,y,G1 implies e DSJoins x,y,G2)
by Def32;
