reserve k,n,i,j for Nat;

theorem
  for K being Field, M being Matrix of 2,K holds Det M = (M*(1,1))*(M*(2
  ,2))-(M*(1,2))*(M*(2,1))
proof
  reconsider s1=<*1,2*>, s2=<*2,1*> as Permutation of Seg 2 by Th2;
  let K be Field, M be Matrix of 2,K;
A1: now
A2: s1.1=1;
    assume s1=s2;
    hence contradiction by A2,FINSEQ_1:44;
  end;
  set D0={s1,s2};
  reconsider l0=<*>D0 as FinSequence of Group_of_Perm(2) by Th3,MATRIX_1:def 13
;
  set X=Permutations 2;
  reconsider p1 = s1, p2 = s2 as Element of Permutations(2) by MATRIX_1:def 12;
  set Y=the carrier of K;
  set f=Path_product M;
  set F=the addF of K;
  set di=(the multF of K) $$ (Path_matrix(p1,M));
  set B=In (Permutations 2,Fin Permutations 2);
  Permutations 2 in Fin Permutations 2 by FINSUB_1:def 5; then
A3: B=Permutations 2 by SUBSET_1:def 8;
  Det M=(the addF of K) $$ (In(Permutations 2,Fin Permutations 2)
,Path_product M) by
MATRIX_3:def 9;
  then consider G being Function of Fin X, Y such that
A4: Det M = G.B and
  for e being Element of Y st e is_a_unity_wrt F holds G.{} = e and
A5: for x being Element of X holds G.{x} = f.x and
A6: for B9 being Element of Fin X st B9 c= B & B9 <> {} for x being
  Element of X st x in B \ B9 holds G.(B9 \/ {x}) = F.(G.B9,f.x) by A3,
SETWISEO:def 3;
A7: G.{p1} = f.p1 by A5;
A8: G.B = (the addF of K).(f.p1,f.p2)
  proof
    reconsider B9={.p1.} as Element of Fin X;
A9: B9 c= B
    proof
      let y be object;
      assume y in B9;
      then y=p1 by TARSKI:def 1;
      hence thesis by A3;
    end;
    B\B9 ={s2} by A1,A3,Th3,ZFMISC_1:17;
    then s2 in B\B9 by TARSKI:def 1;
    then G.(B9 \/ {p2}) = F.(G.B9,f.p2) by A6,A9;
    hence thesis by A3,A7,Th3,ENUMSET1:1;
  end;
  set dj=(the multF of K) $$ (Path_matrix(p2,M));
A10: p1.2 = 2;
A11: p2.2 = 1;
A12: len Path_matrix(p1,M) = 2 by MATRIX_3:def 7;
  then consider f3 being sequence of the carrier of K such that
A13: f3.1 = (Path_matrix(p1,M)).1 and
A14: for n being Nat st 0 <> n & n < 2 holds f3.(n + 1) = (
  the multF of K).(f3.n,(Path_matrix(p1,M)).(n + 1)) and
A15: di = f3.2 by FINSOP_1:def 1;
A16: 1 in Seg 2;
  then
A17: 1 in dom Path_matrix(p1,M) by A12,FINSEQ_1:def 3;
A18: 2 in Seg 2;
  then 2 in dom Path_matrix(p1,M) by A12,FINSEQ_1:def 3;
  then
A19: p1.1 = 1 & Path_matrix(p1,M).2=M*(2,2) by A10,MATRIX_3:def 7;
A20: len Path_matrix(p2,M) = 2 by MATRIX_3:def 7;
  then consider f4 being sequence of the carrier of K such that
A21: f4.1 = (Path_matrix(p2,M)).1 and
A22: for n being Nat st 0 <> n & n < 2 holds f4.(n + 1) = (
  the multF of K).(f4.n,(Path_matrix(p2,M)).(n + 1)) and
A23: dj = f4.2 by FINSOP_1:def 1;
A24: 1 in dom Path_matrix(p2,M) by A20,A16,FINSEQ_1:def 3;
  2 in dom Path_matrix(p2,M) by A20,A18,FINSEQ_1:def 3;
  then
A25: p2.1 = 2 & Path_matrix(p2,M).2=M*(2,1) by A11,MATRIX_3:def 7;
A26: f4.(1+1)=(the multF of K).(f4.1,(Path_matrix(p2,M)).(1+1)) by A22
    .=(M*(1,2))*(M*(2,1)) by A24,A25,A21,MATRIX_3:def 7;
A27: len Permutations 2 = 2 by MATRIX_1:9;
  not ex l be FinSequence of Group_of_Perm 2 st (len l)mod 2=0 & s2=
Product l & for i st i in dom l ex q being Element of Permutations 2 st l.i=q &
  q is being_transposition by Lm1,Th11;
  then f.p2 = -((the multF of K) "**" (Path_matrix(p2,M)),p2) & p2 is odd by
A27,MATRIX_3:def 8;
  then
A28: f.p2 = - dj by MATRIX_1:def 16;
A29: Product l0=Product <*> (the carrier of Group_of_Perm 2)
    .= 1_(Group_of_Perm(2)) by GROUP_4:8
    .= p1 by FINSEQ_2:52,MATRIX_1:15;
A30: 0 mod 2=0 by NAT_D:26;
  ex l be FinSequence of Group_of_Perm 2 st (len l)mod 2=0 & s1=Product l
  & for i st i in dom l ex q being Element of Permutations 2 st l.i=q & q is
  being_transposition
  proof
    take l0;
    thus (len l0)mod 2=0 & s1=Product l0 by A30,A29;
    let i;
    thus thesis;
  end;
  then
A31: f.p1 = -((the multF of K) "**" (Path_matrix(p1,M)),p1) & p1 is even by A27
,MATRIX_3:def 8;
  f3.(1+1)=(the multF of K).(f3.1,(Path_matrix(p1,M)).(1+1)) by A14
    .=(M*(1,1))*(M*(2,2)) by A17,A19,A13,MATRIX_3:def 7;
  hence thesis by A4,A8,A15,A23,A26,A31,A28,MATRIX_1:def 16;
end;
