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
  for A be Matrix of len b1,len B2,K holds 
  a * Mx2Tran(A,b1,B2) = Mx2Tran(a * A,b1,B2)
proof
  let A be Matrix of len b1,len B2,K;
  set aA=a*A;
  set aM=Mx2Tran(aA,b1,B2);
  set M=Mx2Tran(A,b1,B2);
  now
    let x be object;
    assume x in the carrier of V1;
    then reconsider v=x as Element of V1;
    set L=LineVec2Mx(v|--b1);
    set amA=lmlt(a*Line(L*A,1),B2);
    set mA=lmlt(Line(L*A,1),B2);
A1: width L=len (v|--b1) & len (v|--b1)=len b1 by MATRIX_0:23,MATRLIN:def 7;
A2: len A=len b1 by MATRIX_0:def 2;
    len L=1 by MATRIX_0:23;
    then
A3: len (L*A)=1 by A1,A2,MATRIX_3:def 4;
A4: dom mA=dom (Line(L*A,1))/\dom B2 by Lm1;
    len (a*Line(L*A,1))=len Line(L*A,1) by MATRIXR1:16;
    then
A5: dom (a*Line(L*A,1))=dom Line(L*A,1) by FINSEQ_3:29;
A6: dom amA=dom (a*Line(L*A,1))/\dom B2 by Lm1;
    then
A7: len mA=len amA by A5,A4,FINSEQ_3:29;
A8: now
      let k be Nat such that
A9:   k in dom mA;
A10:  mA.k=mA/.k by A9,PARTFUN1:def 6;
      k in dom Line(L*A,1) by A4,A9,XBOOLE_0:def 4;
      then
A11:  Line(L*A,1).k=Line(L*A,1)/.k by PARTFUN1:def 6;
      k in dom B2 by A4,A9,XBOOLE_0:def 4;
      then
A12:  B2.k=B2/.k by PARTFUN1:def 6;
      k in dom (a*Line(L*A,1)) by A5,A4,A9,XBOOLE_0:def 4;
      then (a*Line(L*A,1)).k=a*(Line(L*A,1)/.k) by A11,FVSUM_1:50;
      hence amA.k = (a*(Line(L*A,1)/.k))*B2/.k by A6,A5,A4,A9,A12,FUNCOP_1:22
        .= a*((Line(L*A,1)/.k)*B2/.k) by VECTSP_1:def 16
        .= a*(mA/.k) by A9,A11,A12,A10,FUNCOP_1:22;
    end;
    thus aM.x = Sum lmlt (Line(L * aA,1),B2) by Def3
      .= Sum lmlt (Line(a*(L*A),1),B2) by A1,A2,MATRIXR1:22
      .= Sum amA by A3,MATRIXR1:20
      .= a*Sum mA by A7,A8,RLVECT_2:67
      .= a*M.v by Def3
      .= (a*M).x by MATRLIN:def 4;
  end;
  hence thesis by FUNCT_2:12;
end;
