 reserve X,Y for set,
         n,m,k,i for Nat,
         r for Real,
         R for Element of F_Real,
         K for Field,
         f,f1,f2,g1,g2 for FinSequence,
         rf,rf1,rf2 for real-valued FinSequence,
         cf,cf1,cf2 for complex-valued FinSequence,
         F for Function;
reserve f,f1,f2 for n-element real-valued FinSequence,
        p,p1,p2 for Point of TOP-REAL n,
        M,M1,M2 for Matrix of n,m,F_Real,
        A,B for Matrix of n,F_Real;

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;
