theorem
  for i, j st i in Seg n & j in Seg n & i <> j holds Det A = Det RLine(A
  ,i,Line(A,i) + a*Line(A,j))
proof
  let i, j such that
A1: i in Seg n and
A2: j in Seg n and
A3: i <> j;
A4: width A = n by MATRIX_0:24;
  then
A5: len Line(A,j)=n by MATRIX_0:def 7;
A6: len Line(A,j)=len (a*Line(A,j)) by Lm5;
  len Line(A,i)=n by A4,MATRIX_0:def 7;
  hence Det RLine(A,i,Line(A,i)+a*Line(A,j))= Det(RLine(A,i,Line(A,i))) + Det(
  RLine(A,i,a*Line(A,j))) by A1,A5,A6,Th36
    .=Det(A)+Det(RLine(A,i,a*Line(A,j))) by Th30
    .=Det(A)+0.K by A1,A2,A3,Th52
    .=Det(A) by RLVECT_1:4;
end;
