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 Th118:
  for G1 being _Graph, G2 being spanning Subgraph of G1
  st the_Edges_of G1 = the_Edges_of G2 holds G1 == G2
proof
  let G1 be _Graph, G2 be spanning Subgraph of G1;
  assume A1: the_Edges_of G1 = the_Edges_of G2;
  A2: G1 is Subgraph of G1 by Lm3;
  the_Vertices_of G1 = the_Vertices_of G2 by Def33;
  hence thesis by A1, A2, Th86;
end;
