
theorem Th78:
  for G1 being _Graph, G2 being Subgraph of G1, E being RepEdgeSelection of G1
  st E c= the_Edges_of G2 holds E is RepEdgeSelection of G2
proof
  let G1 be _Graph, G2 be Subgraph of G1, E be RepEdgeSelection of G1;
  assume A1: E c= the_Edges_of G2;
  now
    let v,w,e0 be object;
    A2: v is set & w is set by TARSKI:1;
    assume e0 Joins v,w,G2;
    then consider e being object such that
      A3: e Joins v,w,G1 & e in E and
      A4: for e9 being object st e9 Joins v,w,G1 & e9 in E holds e9 = e
      by A2, Def5, GLIB_000:72;
    take e;
    thus e Joins v,w,G2 & e in E by A1, A2, A3, GLIB_000:73;
    let e9 be object;
    assume e9 Joins v,w,G2 & e9 in E;
    hence e9 = e by A2, A4, GLIB_000:72;
  end;
  hence thesis by A1, Def5;
end;
