
theorem Th52:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st F is strong_SG-embedding holds G2 is complete implies G1 is complete
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  assume A1: F is strong_SG-embedding;
  assume A2: G2 is complete;
  now
    let v,w be Vertex of G1;
    assume A3: v <> w;
    v in the_Vertices_of G1 & w in the_Vertices_of G1;
    then A4: v in dom F_V & w in dom F_V by A1, Def11;
    then F_V.v in rng F_V & F_V.w in rng F_V by FUNCT_1:3;
    then reconsider v2 = F_V.v, w2 = F_V.w as Vertex of G2;
    v2 <> w2 by A1, A3, A4, FUNCT_1:def 4;
    then consider e2 being object such that
      A5: e2 Joins v2,w2,G2 by A2, CHORD:def 6, CHORD:def 3;
    consider e1 being object such that
      A6: e1 Joins v,w,G1 & e1 in dom F_E & F_E.e1 = e2 by A1, A4, A5, Def16;
    thus v,w are_adjacent by A6, CHORD:def 3;
  end;
  hence thesis by CHORD:def 6;
end;
