
theorem Th1:
  for c being non empty Cardinal, v being Vertex of canCompleteGraph(c)
  holds v.inNeighbors() = v & v.outNeighbors() = c \ succ v
proof
  let c be non empty Cardinal, v be Vertex of canCompleteGraph(c);
  now
    let x be object;
    hereby
      assume A1: x in v.inNeighbors();
      then reconsider a = x as Ordinal;
      consider e being object such that
        A2: e DJoins x,v,canCompleteGraph(c) by A1, GLIB_000:69;
      e = [x,v] by A2, GLUNIR00:64;
      then [x,v] in RelIncl c \ id c by A2, GLUNIR00:63;
      then [x,v] in RelIncl c by XBOOLE_0:def 5;
      then A3: a c= v by ORDINAL6:8;
      x <> v by A2, GLIB_000:136;
      then not v c= a by A3, XBOOLE_0:def 10;
      then a in v by ORDINAL1:16;
      hence x in v;
    end;
    assume A4: x in v;
    then reconsider a = x as Vertex of canCompleteGraph(c) by ORDINAL1:10;
    a <> v
    proof
      assume a = v;
      then v in v by A4;
      hence contradiction;
    end;
    then a c= v & a <> v by A4, ORDINAL1:def 2;
    then [a,v] in RelIncl c & not [a,v] in id c
      by WELLORD2:def 1, RELAT_1:def 10;
    then [a,v] in RelIncl c \ id c by XBOOLE_0:def 5;
    hence x in v.inNeighbors() by GLIB_000:69, GLUNIR00:63;
  end;
  hence v.inNeighbors() = v by TARSKI:2;
  now
    let x be object;
    hereby
      assume A5: x in v.outNeighbors();
      then reconsider a = x as Vertex of canCompleteGraph(c);
      consider e being object such that
        A6: e DJoins v,x,canCompleteGraph(c) by A5, GLIB_000:70;
      e = [v,x] by A6, GLUNIR00:64;
      then [v,x] in RelIncl c \ id c by A6, GLUNIR00:63;
      then [v,x] in RelIncl c by XBOOLE_0:def 5;
      then A7: v c= a by ORDINAL6:8;
      x <> v by A6, GLIB_000:136;
      then not a c= v by A7, XBOOLE_0:def 10;
      then not succ a c= succ v by ORDINAL2:1;
      then not a in succ v by ORDINAL1:21;
      hence x in c \ succ v by XBOOLE_0:def 5;
    end;
    assume x in c \ succ v;
    then A8: x in c & not x in succ v by XBOOLE_0:def 5;
    then reconsider a = x as Vertex of canCompleteGraph(c);
    not succ a c= succ v by A8, ORDINAL1:21;
    then A9: v in a & v <> a by ORDINAL1:16, ORDINAL2:1;
    then v c= a by ORDINAL1:def 2;
    then [v,a] in RelIncl c & not [v,a] in id c
      by A9, WELLORD2:def 1, RELAT_1:def 10;
    then [v,a] in RelIncl c \ id c by XBOOLE_0:def 5;
    hence x in v.outNeighbors() by GLIB_000:70, GLUNIR00:63;
  end;
  hence thesis by TARSKI:2;
end;
