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 Th48:
  for i,n being Element of NAT for p being Element of Permutations
  n st i in Seg n holds ex k being Element of NAT st k in Seg n & i = p.k
proof
  let i,n be Element of NAT;
  let p be Element of Permutations n;
A1: p is Permutation of Seg n by MATRIX_1:def 12;
  then
A2: dom p = Seg n by FUNCT_2:52;
  then reconsider s = p as FinSequence by FINSEQ_1:def 2;
  assume i in Seg n;
  then i in rng s by A1,FUNCT_2:def 3;
  then ex k being Nat st k in dom s & i = s.k by FINSEQ_2:10;
  hence thesis by A2;
end;
