theorem Th18:
  1 <= i & i <= m & n <> 0 implies (Mx2Tran M).f.i = @f"*"Col(M,i)
proof
  assume that
   A1: 1<=i & i<=m and
   A2: n<>0;
  A3: len M=n by A2,MATRIX13:1;
  set Lf=LineVec2Mx(@f);
  set LfM=Lf*M;
  len f=n by CARD_1:def 7;
  then A4: width Lf=n by MATRIX13:1;
  width M=m by A2,MATRIX13:1;
  then A5: width LfM=m by A4,A3,MATRIX_3:def 4;
  len Lf=1 by MATRIX13:1;
  then len LfM=1 by A4,A3,MATRIX_3:def 4;
  then A6: [1,i] in Indices LfM by A1,A5,MATRIX_0:30;
  set LM=Line(LfM,1);
  i in Seg m & (Mx2Tran M).f=LM by A1,A2,Def3;
  hence (Mx2Tran M).f.i=LfM*(1,i) by A5,MATRIX_0:def 7
   .=Line(Lf,1)"*"Col(M,i) by A4,A3,A6,MATRIX_3:def 4
   .=@f"*"Col(M,i) by MATRIX15:25;
end;
