reserve G for _Graph;

theorem
  for G being loopless _Graph st VertexAdjSymRel(G) is total
  holds for C being Component of G holds C is non _trivial
proof
  let G be loopless _Graph;
  assume VertexAdjSymRel(G) is total;
  then A1: dom VertexAdjSymRel(G) = the_Vertices_of G by PARTFUN1:def 2;
  field VertexDomRel(G)
     = field VertexDomRel(G) \/ field VertexDomRel(G)
    .= field VertexDomRel(G) \/ field ((VertexDomRel(G))~) by RELAT_1:21
    .= field VertexAdjSymRel(G) by RELAT_1:18
    .= dom VertexAdjSymRel(G) \/ rng VertexAdjSymRel(G) by RELAT_1:def 6;
  then A2: the_Vertices_of G c= field VertexDomRel(G) by A1, XBOOLE_1:7;
  field VertexDomRel(G) c= the_Vertices_of G \/the_Vertices_of G by RELSET_1:8;
  hence thesis by A2, Th5, XBOOLE_0:def 10;
end;
