
theorem
  for G being _Graph, V1, V2 being Subset of the_Vertices_of G
  for G1 being addLoops of G, V1, G2 being addLoops of G1, V2
  st V1 misses V2 holds G2 is addLoops of G, V1 \/ V2
proof
  let G be _Graph, V1, V2 being Subset of the_Vertices_of G;
  let G1 be addLoops of G, V1, G2 be addLoops of G1, V2;
  assume A1: V1 misses V2;
  consider E1 being set, f1 being one-to-one Function such that
    A2: E1 misses the_Edges_of G & the_Edges_of G1 = the_Edges_of G \/ E1 &
      dom f1 = E1 & rng f1 = V1 & the_Source_of G1 = the_Source_of G +* f1 &
      the_Target_of G1 = the_Target_of G +* f1 by Def5;
  V2 c= the_Vertices_of G;
  then V2 c= the_Vertices_of G1 by Th15;
  then consider E2 being set, f2 being one-to-one Function such that
    A3: E2 misses the_Edges_of G1 & the_Edges_of G2 = the_Edges_of G1 \/ E2 &
      dom f2 = E2 & rng f2 = V2 & the_Source_of G2 = the_Source_of G1 +* f2 &
      the_Target_of G2 = the_Target_of G1 +* f2 by Def5;
  A4: G2 is Supergraph of G by GLIB_006:62;
  A5: the_Vertices_of G2 = the_Vertices_of G1 by Th15
    .= the_Vertices_of G by Th15;
  now
    reconsider E = E1 \/ E2 as set;
    reconsider f = f1 +* f2 as one-to-one Function by A1, A2, A3, FUNCT_4:92;
    take E,f;
    A6: E2 misses the_Edges_of G & E2 misses E1 by A2, A3, XBOOLE_1:70;
    hence E misses the_Edges_of G by A2, XBOOLE_1:70;
    thus the_Edges_of G2 =  the_Edges_of G \/ E by A2, A3, XBOOLE_1:4;
    thus dom f = E by A2, A3, FUNCT_4:def 1;
    thus rng f = V1 \/ V2 by A2, A3, A6, NECKLACE:6;
    thus the_Source_of G2 = the_Source_of G +* f by A2, A3, FUNCT_4:14;
    thus the_Target_of G2 = the_Target_of G +* f by A2, A3, FUNCT_4:14;
  end;
  hence thesis by A4, A5, Def5;
end;
