reserve X for set;

theorem Th34:
  ex ee being Subset of TWOELEMENTSETS(Seg 3) st
  ee = {.{.1,2.},{.2,3.},{.3,1.}.} &
  TriangleGraph = SimpleGraphStruct (#(Seg 3),ee#)
proof
  consider ee being finite Subset of TWOELEMENTSETS(Seg 3) such that
A1: ee = {{i,j} where i,j is Element of NAT : i in (Seg 3) & j in (Seg 3
  ) & i<>j} and
A2: TriangleGraph = SimpleGraphStruct (#(Seg 3),ee#) by Def13;
  take ee;
  now
    let a be object;
    assume a in ee;
    then consider i,j being Element of NAT such that
A3: a={i,j} and
A4: i in (Seg 3) and
A5: j in (Seg 3) and
A6: i<>j by A1;
    per cases by A4,ENUMSET1:def 1,FINSEQ_3:1;
    suppose
A7:   i=1;
      now
        per cases by A5,ENUMSET1:def 1,FINSEQ_3:1;
        suppose
          j=1;
          hence a in {.{.1,2.},{.2,3.},{.3,1.}.} by A6,A7;
        end;
        suppose
          j=2;
          hence a in {.{.1,2.},{.2,3.},{.3,1.}.} by A3,A7,ENUMSET1:def 1;
        end;
        suppose
          j=3;
          hence a in {.{.1,2.},{.2,3.},{.3,1.}.} by A3,A7,ENUMSET1:def 1;
        end;
      end;
      hence a in {.{.1,2.},{.2,3.},{.3,1.}.};
    end;
    suppose
A8:   i=2;
      now
        per cases by A5,ENUMSET1:def 1,FINSEQ_3:1;
        suppose
          j=1;
          hence a in {.{.1,2.},{.2,3.},{.3,1.}.} by A3,A8,ENUMSET1:def 1;
        end;
        suppose
          j=2;
          hence a in {.{.1,2.},{.2,3.},{.3,1.}.} by A6,A8;
        end;
        suppose
          j=3;
          hence a in {.{.1,2.},{.2,3.},{.3,1.}.} by A3,A8,ENUMSET1:def 1;
        end;
      end;
      hence a in {.{.1,2.},{.2,3.},{.3,1.}.};
    end;
    suppose
A9:   i=3;
      now
        per cases by A5,ENUMSET1:def 1,FINSEQ_3:1;
        suppose
          j=1;
          hence a in {.{.1,2.},{.2,3.},{.3,1.}.} by A3,A9,ENUMSET1:def 1;
        end;
        suppose
          j=2;
          hence a in {.{.1,2.},{.2,3.},{.3,1.}.} by A3,A9,ENUMSET1:def 1;
        end;
        suppose
          j=3;
          hence a in {.{.1,2.},{.2,3.},{.3,1.}.} by A6,A9;
        end;
      end;
      hence a in {.{.1,2.},{.2,3.},{.3,1.}.};
    end;
  end;
  then
A10: ee c= {.{.1,2.},{.2,3.},{.3,1.}.};
  now
    let e be object;
    assume
A11: e in {.{.1,2.},{.2,3.},{.3,1.}.};
    per cases by A11,ENUMSET1:def 1;
    suppose
A12:  e={1,2};
      now
        take i=1,j=2;
        thus e={i,j} by A12;
        thus i in Seg 3 & j in (Seg 3);
        thus i<>j;
      end;
      hence e in ee by A1;
    end;
    suppose
A13:  e={2,3};
      now
        take i=2,j=3;
        thus e={i,j} & i in (Seg 3) & j in (Seg 3) & i<>j by A13;
      end;
      hence e in ee by A1;
    end;
    suppose
A14:  e={3,1};
      now
        take i=3,j=1;
        thus e={i,j} & i in (Seg 3) & j in (Seg 3) & i<>j by A14;
      end;
      hence e in ee by A1;
    end;
  end;
  then {.{.1,2.},{.2,3.},{.3,1.}.} c= ee;
  hence thesis by A2,A10,XBOOLE_0:def 10;
end;
