reserve x,y for object,X,Y for set,
  D for non empty set,
  i,j,k,l,m,n,m9,n9 for Nat,
  i0,j0,n0,m0 for non zero Nat,
  K for Field,
  a,b for Element of K,
  p for FinSequence of K,
  M for Matrix of n,K;
reserve A for (Matrix of D),
  A9 for Matrix of n9,m9,D,
  M9 for Matrix of n9, m9,K,
  nt,nt1,nt2 for Element of n-tuples_on NAT,
  mt,mt1 for Element of m -tuples_on NAT,
  M for Matrix of K;
reserve P,P1,P2,Q,Q1,Q2 for without_zero finite Subset of NAT;

theorem Th71:
  for M be Matrix of n,K holds Det (a*M) = power(K).(a,n) * Det M
proof
  let M be Matrix of n,K;
  defpred P[Nat] means for k st k=$1 & k <=n ex aM be Matrix of n,K st Det aM
= power(K).(a,k) * Det M & for i st i in Seg n holds (i<=k implies Line(aM,i)=a
  * Line(M,i)) & (i > k implies Line(aM,i)=Line(M,i));
A1: for m st P[m] holds P[m+1]
  proof
    let m such that
A2: P[m];
    let k such that
A3: k=m+1 and
A4: k <=n;
    m<=n by A3,A4,NAT_1:13;
    then consider aM be Matrix of n,K such that
A5: Det aM = power(K).(a,m) * Det M and
A6: for i st i in Seg n holds (i<=m implies Line(aM,i)=a*Line(M,i))&
    (i>m implies Line(aM,i)=Line(M,i)) by A2;
    take R=RLine(aM,k,a*Line(aM,k));
    0+1<=k by A3,XREAL_1:7;
    then k in Seg n by A4;
    hence Det R = a * (power(K).(a,m) * Det M) by A5,MATRIX11:35
      .= power(K).(a,m) *a * Det M by GROUP_1:def 3
      .= power(K).(a,k) *Det M by A3,GROUP_1:def 7;
    let i such that
A7: i in Seg n;
    per cases by XXREAL_0:1;
    suppose
A8:   i< k or i>k;
      then
A9:  i<=m & i<k or i>m & i>k by A3,NAT_1:13;
      Line(R,i)=Line(aM,i) by A7,A8,MATRIX11:28;
      hence thesis by A6,A7,A9;
    end;
    suppose
A10:  i=k;
      len (a*Line(aM,k)) = len Line(aM,k) by MATRIXR1:16
        .= width aM by MATRIX_0:def 7;
      then
A11:  Line(R,i)=a*Line(aM,k) by A7,A10,MATRIX11:28;
      i > m by A3,A10,NAT_1:13;
      hence thesis by A6,A7,A10,A11;
    end;
  end;
A12: P[0]
  proof
    let k such that
A13: k=0 and
    k <=n;
    take aM=M;
    power(K).(a,0)=1_K by GROUP_1:def 7;
    hence Det aM = power(K).(a,k) * Det M by A13;
    let i;
    assume i in Seg n;
    hence thesis by A13;
  end;
  for m holds P[m] from NAT_1:sch 2(A12,A1);
  then consider aM be Matrix of n,K such that
A14: Det aM = power(K).(a,n) * Det M and
A15: for i st i in Seg n holds (i<=n implies Line(aM,i)=a*Line(M,i))&(i>
  n implies Line(aM,i)=Line(M,i));
  set AM=a*M;
A16: len AM = n by MATRIX_0:def 2;
A17: len M=n by MATRIX_0:def 2;
A18: now
    let i such that
A19: 1<=i and
A20: i<=n;
A21: i in Seg n by A19,A20;
    hence aM.i = Line(aM,i) by MATRIX_0:52
      .= a*Line(M,i) by A15,A20,A21
      .= Line(AM,i) by A17,A19,A20,MATRIXR1:20
      .= AM.i by A21,MATRIX_0:52;
  end;
  len aM=n by MATRIX_0:def 2;
  hence thesis by A14,A16,A18,FINSEQ_1:14;
end;
