reserve GS for GraphStruct;
reserve G,G1,G2,G3 for _Graph;
reserve e,x,x1,x2,y,y1,y2,E,V,X,Y for set;
reserve n,n1,n2 for Nat;
reserve v,v1,v2 for Vertex of G;

theorem Th106:
  for G being _trivial _Graph ex v being Vertex of G st the_Vertices_of G = {v}
    & the_Source_of G = the_Edges_of G --> v
    & the_Target_of G = the_Edges_of G --> v
proof
  let G be _trivial _Graph;
  consider v being Vertex of G such that
    A1: the_Vertices_of G = {v} by Th22;
  take v;
  thus the_Vertices_of G = {v} by A1;
  for e being object st e in dom the_Source_of G holds (the_Source_of G).e = v
  proof
    let e be object;
    assume e in dom the_Source_of G;
    then (the_Source_of G).e in the_Vertices_of G by PARTFUN1:4;
    hence thesis by A1, TARSKI:def 1;
  end;
  hence the_Source_of G = (dom the_Source_of G) --> v by FUNCOP_1:11
    .= the_Edges_of G --> v by Th4;
  for e being object st e in dom the_Target_of G holds (the_Target_of G).e = v
  proof
    let e be object;
    assume e in dom the_Target_of G;
    then (the_Target_of G).e in the_Vertices_of G by PARTFUN1:4;
    hence thesis by A1, TARSKI:def 1;
  end;
  hence the_Target_of G = (dom the_Target_of G) --> v by FUNCOP_1:11
    .= the_Edges_of G --> v by Th4;
end;
