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

theorem Th10:
  for p being Element of Permutations 2 st p = Rev idseq 2 holds
  Path_matrix (p, (a,b)][(c,d)) = <* b,c *>
proof
  let p be Element of Permutations 2 such that
A1: p = Rev 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 1 in dom Path_matrix (p, (a,b)][(c,d));
  then
A4: Path_matrix (p, (a,b)][(c,d)).1=(a,b)][(c,d)*(1,2) by A1,Th2,MATRIX_3:def 7
    .= b by MATRIX_0:50;
  2 in dom Path_matrix (p, (a,b)][(c,d)) by A3;
  then Path_matrix (p, (a,b)][(c,d)).2=(a,b)][(c,d)*(2,1) by A1,Th2,
MATRIX_3:def 7
    .=c by MATRIX_0:50;
  hence thesis by A2,A4,FINSEQ_1:44;
end;
