
theorem
  for G being _Graph, x,y being set holds x in G.AdjacentSet({y}) iff y
  in G.AdjacentSet({x})
proof
  let G be _Graph, x,y be set;
  hereby
    assume
A1: x in G.AdjacentSet({y});
    then reconsider xg = x as Vertex of G;
    now
      consider vy being Vertex of G such that
A2:   vy in {y} and
      vy,xg are_adjacent by A1,Th49;
      assume
A3:   not y in the_Vertices_of G;
      vy = y by A2,TARSKI:def 1;
      hence contradiction by A3;
    end;
    then reconsider yg = y as Vertex of G;
A4: xg,yg are_adjacent by A1,Th51;
    xg <> yg by A1,Th51;
    hence y in G.AdjacentSet({x}) by A4,Th51;
  end;
  assume
A5: y in G.AdjacentSet({x});
  then reconsider yg = y as Vertex of G;
  now
    consider vx being Vertex of G such that
A6: vx in {x} and
    vx,yg are_adjacent by A5,Th49;
    assume
A7: not x in the_Vertices_of G;
    vx = x by A6,TARSKI:def 1;
    hence contradiction by A7;
  end;
  then reconsider xg = x as Vertex of G;
A8: xg,yg are_adjacent by A5,Th51;
  xg <> yg by A5,Th51;
  hence thesis by A8,Th51;
end;
