
theorem Th19:
  for G being _Graph, S being Subset of the_Vertices_of G
  for H being inducedSubgraph of G,S
  for u,v being object st u in S & v in S
  for e being object st e Joins u,v,G holds e Joins u,v,H
proof
  let G be _Graph, S be Subset of the_Vertices_of G;
  let H be inducedSubgraph of G,S;
  let u,v be object such that
A1: u in S and
A2: v in S;
  reconsider S as non empty Subset of the_Vertices_of G by A1;
  let e be object such that
A3: e Joins u,v,G;
  e in G.edgesBetween(S) by A1,A2,A3,GLIB_000:32;
  then
A4: e in the_Edges_of H by GLIB_000:def 37;
  the_Target_of H = (the_Target_of G) | the_Edges_of H by GLIB_000:45;
  then
A5: (the_Target_of H).e = (the_Target_of G).e by A4,FUNCT_1:49;
A6: (the_Source_of G).e = u & (the_Target_of G).e = v or (the_Source_of G).e
  = v & (the_Target_of G).e = u by A3;
  the_Source_of H = (the_Source_of G) | the_Edges_of H by GLIB_000:45;
  then (the_Source_of H).e = (the_Source_of G).e by A4,FUNCT_1:49;
  hence thesis by A4,A6,A5;
end;
