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

theorem
  for a,b,c,d,e,f,g,h,i being Element of K for M being Matrix of 3,K st
M = <*<*a,b,c*>,<*d,e,f*>,<*g,h,i*>*> holds Det M = a*e*i - c*e*g - a*f*h + b*f
  *g - b*d*i + c*d*h
proof
  reconsider a3 = <*1,3,2*>, a4 = <*2,3,1*>, a5 = <*2,1,3*>, a6 = <*3,1,2*> as
  Element of Permutations 3 by Th27;
  reconsider id3 = idseq 3 as Permutation of Seg 3;
  reconsider Id3 = idseq 3 as Element of Permutations 3 by MATRIX_1:def 12;
A1: id3 is even by MATRIX_1:16;
  reconsider rid3 = Rev idseq 3 as Element of Permutations 3 by Th4;
  let a,b,c,d,e,f,g,h,i be Element of K;
  let M be Matrix of 3,K;
  assume
A2: M = <*<*a,b,c*>,<*d,e,f*>,<*g,h,i*>*>;
  Permutations 3 in Fin Permutations 3 by FINSUB_1:def 5; then
  In(Permutations 3,Fin Permutations 3) = Permutations 3 by SUBSET_1:def 8;
  then reconsider
  X = {Id3,rid3,a3,a4,a5,a6} as Element of Fin Permutations 3 by Th15,Th19,
FINSEQ_2:53;
  reconsider B1 = {.Id3,rid3,a3.}, B2 = {.a4,a5,a6.} as Element of Fin
  Permutations 3;
  set F = the addF of K;
A3: In(Permutations 3,Fin Permutations 3) = X & 
  X = {Id3,rid3,a3} \/ {a4,a5,a6} by Th15,Th19,
ENUMSET1:13,FINSEQ_2:53;
  now
    let x be object;
    assume
A4: x in B1;
    then x=Id3 or x=rid3 or x=a3 by ENUMSET1:def 1;
    then not x in B2 by Lm5,Lm6,Lm7,Lm8,Lm9,Th15,ENUMSET1:def 1,FINSEQ_2:53;
    hence x in B1 \ B2 by A4,XBOOLE_0:def 5;
  end;
  then
A5: B1 c= B1 \ B2 by TARSKI:def 3;
  for x be object st x in B1 \ B2 holds x in B1 by XBOOLE_0:def 5;
  then B1 \ B2 c= B1 by TARSKI:def 3;
  then B1 \ B2 = B1 by A5,XBOOLE_0:def 10;
  then
A6: B1 misses B2 by XBOOLE_1:83;
  set r = Path_product M;
A7: 3 = len Permutations 3 by MATRIX_1:9;
  then Id3 is even by MATRIX_1:16;
  then reconsider
  r1 = r.id3, r2 = r.rid3, r3 = r.a3, r4 = r.a4, r5 = r.a5, r6 = r.
  a6 as Element of K by FUNCT_2:5;
A8: r.id3 = -((the multF of K) $$ Path_matrix (Id3,M),Id3) by MATRIX_3:def 8
    .= (the multF of K) $$ Path_matrix (Id3,M) by A1,A7,MATRIX_1:def 16
    .= (the multF of K) $$ <* a,e,i *> by A2,Th20,FINSEQ_2:53
    .= a*e*i by Th26;
A9: r.a6 = -((the multF of K) $$ Path_matrix (a6,M),a6) by MATRIX_3:def 8
    .= (the multF of K) $$ Path_matrix (a6,M) by A7,Lm14,MATRIX_1:def 16
    .= (the multF of K) $$ <* c,d,h *> by A2,Th25
    .= c*d*h by Th26;
A10: r.a5 = -((the multF of K) $$ Path_matrix (a5,M),a5) by MATRIX_3:def 8
    .= -(the multF of K) $$ Path_matrix (a5,M) by A7,Th43,MATRIX_1:def 16
    .= -(the multF of K) $$ <* b,d,i *> by A2,Th24
    .= -b*d*i by Th26;
A11: r.a4 = -((the multF of K) $$ Path_matrix (a4,M),a4) by MATRIX_3:def 8
    .= (the multF of K) $$ Path_matrix (a4,M) by A7,Lm13,MATRIX_1:def 16
    .= (the multF of K) $$ <* b,f,g *> by A2,Th23
    .= b*f*g by Th26;
A12: r.a3 = -((the multF of K) $$ Path_matrix (a3,M),a3) by MATRIX_3:def 8
    .= -(the multF of K) $$ Path_matrix (a3,M) by A7,Th44,MATRIX_1:def 16
    .= -(the multF of K) $$ <* a,f,h *> by A2,Th22
    .= -a*f*h by Th26;
A13: r.rid3 = -((the multF of K) $$ Path_matrix (rid3,M),rid3) by
MATRIX_3:def 8
    .= -(the multF of K) $$ Path_matrix (rid3,M) by A7,Th15,Th42,
MATRIX_1:def 16
    .= -(the multF of K) $$ <* c,e,g *> by A2,Th15,Th21
    .= -c*e*g by Th26;
A14: F $$(B1,r) = r1 + r2 + r3 & F $$(B2,r) = r4 + r5 + r6 by Lm3,Lm4,Lm10,Lm11
,Th15,FINSEQ_2:53,SETWOP_2:3;
  Det M = F $$ (In(Permutations 3,Fin Permutations 3), r) by MATRIX_3:def 9
    .= F.(F $$ (B1,r),F $$ (B2,r)) by A3,A6,SETWOP_2:4
    .= r1 + r2 + r3 + (r4 + (r5 + r6)) by A14,RLVECT_1:def 3
    .= r1 + r2 + r3 + r4 + (r5 + r6) by RLVECT_1:def 3
    .= a*e*i - c*e*g - a*f*h + b*f*g - b*d*i + c*d*h by A8,A13,A12,A11,A10,A9,
RLVECT_1:def 3;
  hence thesis;
end;
