reserve x,y,z for object,
  i,j,n,m for Nat,
  D for non empty set,
  K for non empty doubleLoopStr,
  s,t for FinSequence,
  a,a1,a2,b1,b2,d for Element of D,
  p, p1,p2,q,r for FinSequence of D,
  F for add-associative right_zeroed
  right_complementable Abelian non empty doubleLoopStr;
reserve A,B for Matrix of n,K;
reserve A,A9,B,B9,C for Matrix of n,F;
reserve i,j,n for Nat,
  K for Field,
  a,b for Element of K;
reserve x,y,x1,x2,y1,y2 for set,
  i,j,k,l,n,m for Nat,
  D for non empty set,
  K for Field,
  s,s2 for FinSequence,
  a,b,c,d for Element of D,
  q,r for FinSequence of D,
  a9,b9 for Element of K;

theorem
  Permutations 1 = {idseq 1}
proof
A1: Permutations 1 c={idseq 1}
  proof
    let p be object;
    assume p in Permutations 1;
    then reconsider q=p as Permutation of Seg 1 by Def12;
A2: dom q=Seg 1 by FUNCT_2:52;
    rng q = Seg 1 & 1 in {1} by FUNCT_2:def 3,TARSKI:def 1;
    then q.1 in Seg 1 by FINSEQ_1:2,FUNCT_2:4;
    then
A3: q.1=1 by FINSEQ_1:2,TARSKI:def 1;
    reconsider q as FinSequence by A2,FINSEQ_1:def 2;
    len q= 1 by A2,FINSEQ_1:def 3;
    then q = idseq 1 by A3,FINSEQ_1:40,FINSEQ_2:50;
    hence thesis by TARSKI:def 1;
  end;
  {idseq 1} c= Permutations(1)
  proof
    let x be object;
    assume x in {idseq 1};
    then x= idseq 1 by TARSKI:def 1;
    hence thesis by Def12;
  end;
  hence thesis by A1,XBOOLE_0:def 10;
end;
