reserve x for set,
  i,j,k,n for Nat,
  K for Field;
reserve a,b,c,d for Element of K;
reserve D for non empty set;

theorem Th19:
  Permutations 3 = {<*1,2,3*>,<*3,2,1*>,<*1,3,2*>,<*2,3,1*>,<*2,1,
  3*>,<*3,1,2*>}
proof
  now
    let x be object;
A1: idseq 3 in Permutations 3 by MATRIX_1:def 12;
A2: <*3,1,2*> in Permutations 3
    proof
      reconsider h = <*1,2*> as FinSequence of NAT;
      <*3*> ^ h = <*3,1,2*> by FINSEQ_1:43;
      hence thesis by A1,Th18,FINSEQ_2:53;
    end;
A3: <*2,3,1*> in Permutations 3
    proof
      reconsider h = <*2,3*> as FinSequence of NAT;
      reconsider f = <*1,2,3*> as one-to-one FinSequence-like Function of Seg
      3,Seg 3 by FINSEQ_2:53;
      f = <*1*> ^ h by FINSEQ_1:43;
      hence thesis by A1,Th18,FINSEQ_2:53;
    end;
A4: <*1,3,2*> in Permutations 3
    proof
      reconsider h = <*2,3*> as FinSequence of NAT;
      reconsider f = <*1,2,3*> as one-to-one FinSequence-like Function of Seg
      3,Seg 3 by FINSEQ_2:53;
      Rev h = <*3,2*> by FINSEQ_5:61;
      then f = <*1*> ^ h & <*1*>^Rev h = <*1,3,2*> by FINSEQ_1:43;
      hence thesis by A1,Th17,FINSEQ_2:53;
    end;
A5: <*2,1,3*> in Permutations 3
    proof
      reconsider h = <*1,2*> as FinSequence of NAT;
      <*3*> ^ h in Permutations 3 by A1,Th18,FINSEQ_2:53;
      then Rev h = <*2,1*> & <*3*> ^ Rev h in Permutations 3 by Th17,
FINSEQ_5:61;
      hence thesis by Th18;
    end;
    assume
    x in {<*1,2,3*>,<*3,2,1*>,<*1,3,2*>,<*2,3,1*>, <*2,1,3*>,<*3,1,2 *>};
    then
    x in {<*1,2,3*>,<*3,2,1*>}\/{<*1,3,2*>,<*2,3,1*>,<*2,1,3*>,<*3,1,2 *>
    } by ENUMSET1:12;
    then
    x in {<*1,2,3*>,<*3,2,1*>} or x in {<*1,3,2*>,<*2,3,1*>,<*2,1,3*>,<*3
    ,1,2*>} by XBOOLE_0:def 3;
    then x in {<*1,2,3*>,<*3,2,1*>} or x in {<*1,3,2*>,<*2,3,1*>} \/ {<*2,1,3
    *>,<*3,1,2*>} by ENUMSET1:5;
    then
A6: x in {<*1,2,3*>,<*3,2,1*>} or x in {<*1,3,2*>,<*2,3,1*>} or x in {<*2
    ,1,3*>,<*3,1,2*>} by XBOOLE_0:def 3;
    Rev idseq 3 in Permutations 3 by Th4;
    hence x in Permutations 3 by A6,A1,A4,A3,A5,A2,Th15,FINSEQ_2:53
,TARSKI:def 2;
  end;
  then
A7: {<*1,2,3*>,<*3,2,1*>,<*1,3,2*>,<*2,3,1*>,<*2,1,3*>,<*3,1,2*>} c=
  Permutations 3 by TARSKI:def 3;
  now
    let p be object;
    assume p in Permutations 3;
    then reconsider q=p as Permutation of Seg 3 by MATRIX_1:def 12;
A8: rng q = Seg 3 by FUNCT_2:def 3;
A9: 3 in Seg 3;
    then q.3 in Seg 3 by A8,FUNCT_2:4;
    then
A10: q.3 = 1 or q.3 = 2 or q.3 = 3 by ENUMSET1:def 1,FINSEQ_3:1;
A11: 2 in Seg 3;
    then q.2 in Seg 3 by A8,FUNCT_2:4;
    then
A12: q.2 = 1 or q.2 = 2 or q.2 = 3 by ENUMSET1:def 1,FINSEQ_3:1;
A13: dom q = Seg 3 by FUNCT_2:52;
A14: q.1=1 & q.2=2 & q.3=3 or q.1=3 & q.2=2 & q.3=1 or q.1=1 & q.2=3 & q.3
=2 or q.1=2 & q.2=3 & q.3=1 or q.1=2 & q.2=1 & q.3=3 or q.1=3 & q.2=1 & q.3=2
implies q=<*1,2,3*> or q=<*3,2,1*> or q=<*1,3,2*> or q=<*2,3,1*> or q=<*2,1,3*>
    or q=<*3,1,2*>
    proof
      reconsider q as FinSequence by A13,FINSEQ_1:def 2;
      len q = 3 by A13,FINSEQ_1:def 3;
      hence thesis by FINSEQ_1:45;
    end;
A15: 1 in Seg 3;
    then q.1 in Seg 3 by A8,FUNCT_2:4;
    then q.1 = 1 or q.1 = 2 or q.1 = 3 by ENUMSET1:def 1,FINSEQ_3:1;
    then
    p = <*1,2,3*> or p = <*3,2,1*> or q = <*1,3,2*> or q = <*2,3,1*> or
    q = <*2,1,3*> or q = <*3,1,2*> by A13,A15,A11,A9,A12,A10,A14,FUNCT_1:def 4;
    then
    p in {<*1,2,3*>,<*3,2,1*>} or q in {<*1,3,2*>,<*2,3,1*>} or
    q in {<*2,1,3*>,<*3,1,2*>} by TARSKI:def 2;
    then
    p in {<*1,2,3*>,<*3,2,1*>} or q in {<*1,3,2*>,<*2,3,1*>}\/ {<*2,1,3*>
    ,<*3,1,2*>} by XBOOLE_0:def 3;
    then
    p in {<*1,2,3*>,<*3,2,1*>} or q in {<*1,3,2*>,<*2,3,1*>,<*2,1,3*>,<*3
    ,1,2*>} by ENUMSET1:5;
    then p in {<*1,2,3*>,<*3,2,1*>} \/ {<*1,3,2*>,<*2,3,1*>,<*2,1,3*>,<*3,1,2
    *>} by XBOOLE_0:def 3;
    hence
    p in {<*1,2,3*>,<*3,2,1*>,<*1,3,2*>,<*2,3,1*>,<*2,1,3*>,<*3,1,2*>}
    by ENUMSET1:12;
  end;
  then
  Permutations 3 c= {<*1,2,3*>,<*3,2,1*>,<*1,3,2*>,<*2,3,1*>,<*2,1,3*>,<*
  3,1,2*>} by TARSKI:def 3;
  hence thesis by A7,XBOOLE_0:def 10;
end;
