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

theorem Th9:
  for p being Element of Permutations 2 st p = idseq 2 holds
  Path_matrix (p, (a,b)][(c,d)) = <* a,d *>
proof
  let p be Element of Permutations 2 such that
A1: p = idseq 2;
A2: len Path_matrix (p, (a,b)][(c,d)) = 2 by MATRIX_3:def 7;
  then
A3: dom Path_matrix (p, (a,b)][(c,d)) = Seg 2 by FINSEQ_1:def 3;
  then
A4: 2 in dom Path_matrix (p, (a,b)][(c,d));
  then 2 = p.2 by A1,A3,FUNCT_1:18;
  then
A5: Path_matrix (p, (a,b)][(c,d)).2=(a,b)][(c,d)*(2,2) by A4,MATRIX_3:def 7
    .= d by MATRIX_0:50;
A6: 1 in dom Path_matrix (p, (a,b)][(c,d)) by A3;
  then 1 = p.1 by A1,A3,FUNCT_1:18;
  then Path_matrix (p, (a,b)][(c,d)).1=(a,b)][(c,d)*(1,1) by A6,MATRIX_3:def 7
    .= a by MATRIX_0:50;
  hence thesis by A2,A5,FINSEQ_1:44;
end;
