 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
  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;
