reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a,a1,a2 for Element of K,
  D for non empty set,
  d,d1,d2 for Element of D,
  M,M1,M2 for (Matrix of D),
  A,A1,A2,B1,B2 for (Matrix of K),
  f,g for FinSequence of NAT;
reserve F,F1,F2 for FinSequence_of_Matrix of D,
  G,G9,G1,G2 for FinSequence_of_Matrix of K;
reserve S,S1,S2 for FinSequence_of_Square-Matrix of D,
  R,R1,R2 for FinSequence_of_Square-Matrix of K;
reserve N for (Matrix of n,K),
  N1 for (Matrix of m,K);

theorem Th49:
  Det <*N*> = <*Det N*>
proof
  set F=<*N*>;
A1: len F=1 by FINSEQ_1:40;
A2: n=len N by MATRIX_0:def 2;
A4: len F=len (Det F) by CARD_1:def 7;
A5: dom (Det F)=Seg len F by FINSEQ_2:124;
  1 in Seg 1;
  then (Det F).1=Det (F.1) by A1,A5,Def7;
  hence thesis by A1,A4,A2,FINSEQ_1:40;
end;
