reserve i, j, m, n, k for Nat,
  x, y for set,
  K for Field,
  a,a1 for Element of K;
reserve V1,V2,V3 for finite-dimensional VectSp of K,
  f for Function of V1,V2,

  b1,b19 for OrdBasis of V1,
  B1 for FinSequence of V1,
  b2 for OrdBasis of V2,
  B2 for FinSequence of V2,

  B3 for FinSequence of V3,
  v1,w1 for Element of V1,
  R,R1,R2 for FinSequence of V1,
  p,p1,p2 for FinSequence of K;

theorem Th32:
  for M be Matrix of len b1,len b2,K st len b1 > 0 holds
  LineVec2Mx(Mx2Tran(M,b1,b2).v1|--b2) = LineVec2Mx(v1|--b1) * M
proof
  set L=LineVec2Mx(v1|--b1);
A1: width L=len (v1|--b1) & len (v1|--b1)=len b1 by MATRIX_0:23,MATRLIN:def 7;
  let M be Matrix of len b1,len b2,K such that
A2: len b1 > 0;
A3: len M=len b1 by A2,MATRIX_0:23;
  set LM=L*M;
  width M=len b2 by A2,MATRIX_0:23;
  then width LM=len b2 by A1,A3,MATRIX_3:def 4;
  then len Line(LM,1)=len b2 by CARD_1:def 7;
  then
A4: Sum lmlt (Line(LM,1),b2) |--b2=Line(LM,1) by MATRLIN:36;
  len L=1 by MATRIX_0:23;
  then len LM=1 by A1,A3,MATRIX_3:def 4;
  hence LM = LineVec2Mx(Sum lmlt (Line(LM,1),b2) |--b2) by A4,MATRIX15:25
    .= LineVec2Mx(Mx2Tran(M,b1,b2).v1|--b2) by Def3;
end;
