reserve X for set;

theorem
  for X being non empty set,g being SimpleGraph of X, v being set st v
  in the carrier of g holds ex VV being finite set st VV=(the carrier of g) &
  degree(g,v)<(card VV)
proof
  let X be non empty set, g be SimpleGraph of X, v be set;
  reconsider VV=the carrier of g as finite set by Th21;
  consider ww being finite set such that
A1: ww={w where w is Element of X : w in VV & {v,w} in the SEdges of g} and
A2: degree(g,v)=card ww by Th28;
  assume
A3: v in (the carrier of g);
A4: now
    assume ww=VV;
    then
A5: ex w being Element of X st v=w & w in VV & {v,w} in (the SEdges of g)
    by A3,A1;
    {v,v}={v} by ENUMSET1:29;
    then consider x,y being object such that
    x in VV and
    y in VV and
A6: x<>y and
A7: {v}={x,y} by A5,Th8;
    v=x by A7,ZFMISC_1:4;
    hence ww<>VV by A6,A7,ZFMISC_1:4;
  end;
  take VV;
  ww c= VV
  proof
    let e be object;
    assume e in ww;
    then ex w being Element of X st e=w & w in VV & {v,w} in (the SEdges of g)
    by A1;
    hence thesis;
  end;
  then ww c< VV by A4,XBOOLE_0:def 8;
  hence thesis by A2,CARD_2:48;
end;
