reserve G for _Graph;

theorem Th38:
  for G being _Graph st for C being Component of G holds C is non _trivial
  holds VertexAdjSymRel(G) is total
proof
  let G be _Graph;
  assume for C being Component of G holds C is non _trivial;
  then A1: field VertexDomRel(G) = the_Vertices_of G by Th6;
  dom VertexAdjSymRel(G)
     = dom VertexDomRel(G) \/ dom ((VertexDomRel(G))~) by XTUPLE_0:23
    .= dom VertexDomRel(G) \/ rng VertexDomRel(G) by RELAT_1:20
    .= the_Vertices_of G by A1, RELAT_1:def 6;
  hence thesis by PARTFUN1:def 2;
end;
