reserve X for set;

theorem
  for g being SimpleGraph of X, v being set, vg being finite set st vg=(
the carrier of g) & v in vg & 1+degree(g,v)=(card vg) holds for w being Element
  of vg st v<>w holds ex e being set st e in (the SEdges of g) & e={v,w}
proof
  let g be SimpleGraph of X, v be set, vg be finite set;
  assume that
A1: vg=(the carrier of g) and
A2: v in vg and
A3: 1+degree(g,v)=(card vg);
  vg is Subset of X by A1,Th21;
  then reconsider X as non empty set by A2;
  let w be Element of vg;
  assume
A4: v<>w;
  take {v,w};
  hereby
    now
      consider ww being finite set such that
A5:   ww={vv where vv is Element of X : vv in vg & {v,vv} in (the
      SEdges of g) } and
A6:   degree(g,v)=(card ww) by A1,Th28;
      reconsider wwv=(ww \/ {v}) as finite set;
      assume
A7:   not {v,w} in (the SEdges of g);
A8:   not v in ww & not w in ww
      proof
        hereby
          assume v in ww;
          then ex vv being Element of X st v=vv & vv in vg & {v,vv} in (the
          SEdges of g) by A5;
          then {v} in (the SEdges of g) by ENUMSET1:29;
          then ex z being Subset of vg st {v}=z & (card z)=2 by A1,Th7;
          hence contradiction by CARD_1:30;
        end;
        assume w in ww;
        then ex vv being Element of X st w=vv & vv in vg & {v,vv} in (the
        SEdges of g) by A5;
        hence contradiction by A7;
      end;
A9:   now
        let e be object;
        assume e in ww;
        then ex vv being Element of X st e=vv & vv in vg & {v,vv} in (the
        SEdges of g) by A5;
        hence e in vg;
      end;
      wwv c= vg & wwv<>vg
      proof
A10:    {v} c= vg by A2,TARSKI:def 1;
        ww c= vg by A9;
        hence wwv c= vg by A10,XBOOLE_1:8;
        assume
A11:    wwv=vg;
        not w in {v} by A4,TARSKI:def 1;
        hence contradiction by A8,A11,XBOOLE_0:def 3;
      end;
      then
A12:  wwv c< vg by XBOOLE_0:def 8;
      (card wwv) = 1+(card ww) by A8,CARD_2:41;
      hence contradiction by A3,A6,A12,CARD_2:48;
    end;
    hence {v,w} in the SEdges of g;
  end;
  thus thesis;
end;
