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

theorem
  Per (a,b)][(c,d) = a * d + b * c
proof
  reconsider rid2 = Rev idseq 2 as Element of Permutations 2 by Th4;
  set M = (a,b)][(c,d);
  reconsider Id2 = idseq 2 as Element of Permutations 2 by MATRIX_1:def 12;
  reconsider id2 = Id2 as Permutation of Seg 2;
  set f = PPath_product M;
A0: 1 in Seg 2;
  Permutations 2 in Fin Permutations 2 by FINSUB_1:def 5; then
A1: In (Permutations 2, Fin Permutations 2) = Permutations 2 &
  id2 <> rid2 by A0,Th2,FUNCT_1:18,SUBSET_1:def 8;
A2: f.rid2 = (the multF of K) $$ Path_matrix (rid2,M) by Def1
    .= (the multF of K) $$ <*b,c*> by Th10
    .= b*c by Th11;
  f.id2 = (the multF of K) $$ Path_matrix (Id2,M) by Def1
    .= (the multF of K) $$ <*a,d*> by Th9
    .= a*d by Th11;
  hence thesis by A2,A1,Th6,SETWOP_2:1;
end;
