reserve x,y,z,x1,x2,y1,y2,z1,z2 for object,
  i,j,k,l,n,m for Nat,
  D for non empty set,
  K for Ring;
reserve a for Element of K;

theorem
  for K be Ring,
      a being Element of K holds
  Det <*<*a*>*> =a
proof
  let K be Ring,
      a be Element of K;
  set M=<*<*a*>*>;
A1: (Path_product M).(idseq 1)=a
  proof
    reconsider p = idseq 1 as Element of Permutations(1) by MATRIX_1:def 12;
A2: -(a,p)=a
    proof
      reconsider q = id Seg 1 as Element of Permutations(1) by MATRIX_1:def 12;
      len Permutations 1 = 1 by MATRIX_1:9;
      then q is even by MATRIX_1:16;
      hence thesis by MATRIX_1:def 16;
    end;
A3: len Path_matrix(p,M) = 1 by Def7;
    then
A4: dom Path_matrix(p,M) = Seg 1 by FINSEQ_1:def 3;
    then
A5: 1 in dom Path_matrix(p,M) by FINSEQ_1:2,TARSKI:def 1;
    then 1=p.1 by A4,FUNCT_1:18;
    then Path_matrix(p,M).1=M*(1,1) by A5,Def7;
    then Path_matrix(p,M).1=a by MATRIX_0:49;
    then
A6: Path_matrix(p,M)=<*a*> by A3,FINSEQ_1:40;
    (Path_product M).p = -((the multF of K) $$ Path_matrix(p,M),p) & <*a*>
    =1|->a by Def8,FINSEQ_2:59;
    hence thesis by A6,A2,FINSOP_1:16;
  end;
  Permutations 1 in Fin Permutations 1 by FINSUB_1:def 5; then
  In (Permutations 1, Fin Permutations 1) = Permutations 1
    by SUBSET_1:def 8; then
  In (Permutations 1, Fin Permutations 1) = {idseq 1} &
    idseq 1 in Permutations 1 by MATRIX_1:10,TARSKI:def 1;
  hence thesis by A1,SETWISEO:17;
end;
