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;
