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 Th27:
  <*1,3,2*> in Permutations 3 & <*2,3,1*> in Permutations 3 & <*2,
  1,3*> in Permutations 3 & <*3,1,2*> in Permutations 3 & <*1,2,3*> in
  Permutations 3 & <*3,2,1*> in Permutations 3
proof
  set h = <*1,2*>;
  reconsider f = <*1,2,3*> as one-to-one FinSequence-like Function of Seg 3,
  Seg 3 by FINSEQ_2:53;
A1: <*3*> ^ h = <*3,1,2*> by FINSEQ_1:43;
A2: <*1,3,2*> in Permutations 3
  proof
    set h = <*2,3*>;
    Rev h = <*3,2*> by FINSEQ_5:61;
    then
A3: <*1*>^Rev h = <*1,3,2*> by FINSEQ_1:43;
    f = <*1*> ^ h & f in Permutations 3 by FINSEQ_1:43,FINSEQ_2:53
,MATRIX_1:def 12;
    hence thesis by A3,Th17;
  end;
A4: idseq 3 in Permutations 3 by MATRIX_1:def 12;
  then <*3*> ^ h in Permutations 3 by Th18,FINSEQ_2:53;
  then
A5: <*3*> ^ Rev h in Permutations 3 by Th17;
  f = <*1*> ^ <*2,3*> & Rev h = <*2,1*> by FINSEQ_1:43,FINSEQ_5:61;
  hence thesis by A4,A2,A5,A1,Th5,Th15,Th18,FINSEQ_2:53;
end;
