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;
