reserve G for _Graph;
reserve G2 for _Graph, G1 for Supergraph of G2;
reserve V for set;
reserve v for object;

theorem Th112:
  for G2 for v1, v2 being Vertex of G2, e being set
  for G1 being addEdge of G2,v1,e,v2 st not e in the_Edges_of G2
  holds G2 is removeEdge of G1,e
proof
  let G2;
  let v1, v2 be Vertex of G2, e be set;
  let G1 be addEdge of G2,v1,e,v2;
  assume A1: not e in the_Edges_of G2;
  then A2: the_Vertices_of G1 = the_Vertices_of G2 &
    the_Edges_of G1 = the_Edges_of G2 \/ {e} &
    the_Source_of G1 = the_Source_of G2 +* (e .--> v1) &
    the_Target_of G1 = the_Target_of G2 +* (e .--> v2) by Def11;
  then A3: the_Edges_of G1 \ {e}
    = the_Edges_of G2 by A1, ZFMISC_1:117;
  the_Vertices_of G1 is non empty & the_Vertices_of G1 c= the_Vertices_of G1;
  then reconsider V = the_Vertices_of G1
    as non empty Subset of the_Vertices_of G1;
  A4: G2 is Subgraph of G1 by Th61;
  the_Edges_of G1 \ {e} c= the_Edges_of G1;
  then the_Edges_of G1 \ {e} c= G1.edgesBetween(V)
    by GLIB_000:34;
  hence thesis by A2, A3, A4, GLIB_000:def 37;
end;
