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
  Det (R|n) = (Det R) |n
proof
A1: len Det R=len R by CARD_1:def 7;
  per cases;
  suppose
A2: n>=len R;
    then R|n=R by FINSEQ_1:58;
    hence thesis by A1,A2,FINSEQ_1:58;
  end;
  suppose
A3: n<len R;
    R=(R|n)^(R/^n) by RFINSEQ:8;
    then
A4: Det R=(Det (R|n))^Det(R/^n) by Th50;
    len (R|n) =n by A3,FINSEQ_1:59;
    then len Det (R|n)=n by CARD_1:def 7;
    hence thesis by A4,FINSEQ_5:23;
  end;
end;
