
theorem Th35:
  for G1, G2 being _Graph
  holds G1 tolerates G2 & the_Vertices_of G1 meets the_Vertices_of G2 iff
    {G1, G2} is /\-tolerating
proof
  let G1, G2 be _Graph;
  hereby
    assume A1: G1 tolerates G2 & the_Vertices_of G1 meets the_Vertices_of G2;
    then the_Vertices_of G1 /\ the_Vertices_of G2 <> {} by XBOOLE_0:def 7;
    then meet {the_Vertices_of G1, the_Vertices_of G2} <> {} by SETFAM_1:11;
    then A2: meet the_Vertices_of {G1, G2} <> {} by Th6;
    {G1, G2} is \/-tolerating by A1, Th19;
    hence {G1, G2} is /\-tolerating by A2;
  end;
  assume A3: {G1, G2} is /\-tolerating;
  hence G1 tolerates G2 by Th19;
  meet {the_Vertices_of G1, the_Vertices_of G2} <> {} by A3, Th6;
  then the_Vertices_of G1 /\ the_Vertices_of G2 <> {} by SETFAM_1:11;
  hence the_Vertices_of G1 meets the_Vertices_of G2 by XBOOLE_0:def 7;
end;
