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, e being set, G2 being removeEdge of G1,e
  holds G1.order() = G2.order() & (e in the_Edges_of G1 implies G2.size() + 1 =
  G1.size())
proof
  let G1 be _finite _Graph, e be set, G2 be removeEdge of G1,e;
A1: the_Edges_of G2 = the_Edges_of G1 \ {e} by Th51;
  thus G1.order() = G2.order() by Th51;
  assume e in the_Edges_of G1;
  then for x being object st x in {e} holds x in the_Edges_of G1
         by TARSKI:def 1;
  then {e} c= the_Edges_of G1;
  then
A2: the_Edges_of G1 = the_Edges_of G2 \/ {e} by A1,XBOOLE_1:45;
  e in {e} by TARSKI:def 1;
  then not e in the_Edges_of G2 by A1,XBOOLE_0:def 5;
  hence thesis by A2,CARD_2:41;
end;
