theorem
  r = R implies r * (Mx2Tran M).f = (Mx2Tran(R*M)).f
proof
  set L=LineVec2Mx@f;
  set RM=R*M;
  set T=Mx2Tran M;
  set TR=Mx2Tran RM;
  assume
   A1: r=R;
  per cases;
  suppose A2: n<>0;
   A3: len M=n by A2,MATRIX13:1;
   len f=n by CARD_1:def 7;
   then A4: width L=n by MATRIX13:1;
   len L=1 by MATRIX13:1;
   then A5: len(L*M)=1 by A4,A3,MATRIX_3:def 4;
   T.f=Line(L*M,1) by A2,Def3;
   hence r*(T.f)=R*Line(L*M,1) by A1,MATRIXR1:17
    .=Line(R*(L*M),1) by A5,MATRIXR1:20
    .=Line(L*RM,1) by A4,A3,MATRIXR1:22
    .=TR.f by A2,Def3;
  end;
  suppose A6: n=0;
    A7: 0.TOP-REAL m = 0* m by EUCLID:70 .= m |-> 0;
    hence r * T.f = r * (m |-> zz) by A6,Def3
    .= m |-> (r*zz) by RVSUM_1:48
    .= TR.f by A6,A7,Def3;
  end;
end;
