theorem Th52:
  for i, j st i in Seg n & j in Seg n & i <> j holds Det RLine(A,i
  ,a*Line(A,j)) = 0.K
proof
  let i, j such that
A1: i in Seg n and
A2: j in Seg n and
A3: i <> j;
  width A = n by MATRIX_0:24;
  then len Line(A,j)=n by MATRIX_0:def 7;
  hence Det(RLine(A,i,a*Line(A,j))) = a*Det(RLine(A,i,Line(A,j))) by A1,Th34
    .= a*0.K by A1,A2,A3,Th51
    .= 0.K;
end;
