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 _finite _Graph, G2 being spanning Subgraph of G1
  st G1.size() = G2.size() holds G1 == G2
proof
  let G1 be _finite _Graph, G2 be spanning Subgraph of G1;
  assume A1: G1.size() = G2.size();
  card the_Edges_of G1 = card the_Edges_of G2 by A1;
  then the_Edges_of G1 = the_Edges_of G2 by CARD_2:102;
  hence thesis by Th118;
end;
