 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 Th23:
  (Mx2Tran M).(r*f) = r * ((Mx2Tran M).f)
proof
  set rf=r*f;
  set T=Mx2Tran M;
  per cases;
  suppose A1: n<>0;
   per cases;
   suppose A2: m=0;
    then T.rf=0.REAL 0;
    hence thesis by A2;
   end;
   suppose m>0;
    reconsider R=r as Element of F_Real by XREAL_0:def 1;
    set Lr=LineVec2Mx@rf;
    set L=LineVec2Mx@f;
    A3: R*@f=@rf by MATRIXR1:17;
    len f=n by CARD_1:def 7;
    then A4: width L=n by MATRIX13:1;
    A5: len M=n by A1,MATRIX13:1;
    len L=1 by MATRIX13:1;
    then A6: len(L*M)=1 by A4,A5,MATRIX_3:def 4;
    T.@f=Line(L*M,1) by A1,Def3;
    hence r*(T.f)=R*Line(L*M,1) by MATRIXR1:17
     .=Line(R*(L*M),1) by A6,MATRIXR1:20
     .=Line((R*L)*M,1) by A4,A5,MATRIX15:1
     .=Line(Lr*M,1) by A3,MATRIX15:29
     .=T.(r*f) by A1,Def3;
   end;
  end;
  suppose A7: n = 0;
    A8: 0.TOP-REAL m = 0* m by EUCLID:70 .= m |-> 0;
    hence T.rf = m |-> (r*zz) by A7,Def3
    .= r * (m |-> zz) by RVSUM_1:48
    .= r * T.f by A7,A8,Def3;
  end;
end;
