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 Th50:
  Det (R1^R2) = Det R1^Det R2
proof
  set R12=R1^R2;
A1: len R12=len R1+len R2 by FINSEQ_1:22;
A2: len (Det R1^Det R2)=len R1+len R2 by CARD_1:def 7;
  len R2=len Det R2 by CARD_1:def 7;
  then
A3: dom (Det R2)=dom R2 by FINSEQ_3:29;
A4: len R1=len Det R1 by CARD_1:def 7;
  then
A5: dom (Det R1)=dom R1 by FINSEQ_3:29;
A6: len Det R12=len R12 by CARD_1:def 7;
  then
A7: dom (Det R1 ^ Det R2)= dom Det R12 by A1,A2,FINSEQ_3:29;
  now
    let k such that
A8: 1<=k and
A9: k<=len R1+len R2;
A10: k in dom (Det R1 ^ Det R2) by A2,A8,A9,FINSEQ_3:25;
    now
      per cases by A10,FINSEQ_1:25;
      suppose
A11:    k in dom (Det R1);
        then
A12:    len (R1.k)=len (R12.k) by A5,FINSEQ_1:def 7;
        thus (Det R1 ^ Det R2).k = (Det R1).k by A11,FINSEQ_1:def 7
          .= Det (R1.k) by A11,Def7
          .= Det(R12.k) by A5,A11,A12,FINSEQ_1:def 7
          .= (Det R12).k by A7,A10,Def7;
      end;
      suppose
        ex n st n in dom (Det R2) & k=len (Det R1)+n;
        then consider n such that
A13:    n in dom (Det R2) and
A14:    k=len R1+n by A4;
A15:    len (R2.n)=len (R12.k) by A3,A13,A14,FINSEQ_1:def 7;
        thus (Det R1 ^ Det R2).k = (Det R2).n by A4,A13,A14,FINSEQ_1:def 7
          .= Det (R2.n) by A13,Def7
          .= Det(R12.k) by A3,A13,A14,A15,FINSEQ_1:def 7
          .= (Det R12).k by A7,A10,Def7;
      end;
    end;
    hence (Det R12).k=(Det R1 ^ Det R2).k;
  end;
  hence thesis by A6,A1,A2;
end;
