
theorem Th31: :: Subgraph02
  for G being _Graph,
      S,T being non empty Subset of the_Vertices_of G st T c= S
  for G2 being inducedSubgraph of G,S holds
    G2.edgesBetween(T) = G.edgesBetween(T)
proof
  let G be _Graph;
  let S,T be non empty Subset of the_Vertices_of G such that
A1: T c= S;
  let G2 be inducedSubgraph of G,S;
A2: the_Edges_of G2 = G.edgesBetween(S) by GLIB_000:def 37;
  now
    let e be object;
    hereby
      assume
A3:   e in G.edgesBetween(T);
      then
A4:   (the_Source_of G).e in T by GLIB_000:31;
A5:   (the_Target_of G).e in T by A3,GLIB_000:31;
A6:   G.edgesBetween(T) c= G.edgesBetween(S) by A1,GLIB_000:36;
      then
A7:   (the_Target_of G2).e = (the_Target_of G).e by A2,A3,GLIB_000:def 32;
      (the_Source_of G2).e = (the_Source_of G).e by A2,A3,A6,GLIB_000:def 32;
      hence e in G2.edgesBetween(T) by A2,A3,A6,A4,A5,A7,GLIB_000:31;
    end;
    assume
A8: e in G2.edgesBetween(T);
    then (the_Source_of G2).e in T by GLIB_000:31;
    then
A9: (the_Source_of G).e in T by A8,GLIB_000:def 32;
    (the_Target_of G2).e in T by A8,GLIB_000:31;
    then
A10: (the_Target_of G).e in T by A8,GLIB_000:def 32;
    e in the_Edges_of G2 by A8;
    hence e in G.edgesBetween(T) by A9,A10,GLIB_000:31;
  end;
  hence thesis by TARSKI:2;
end;
