 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
  (Mx2Tran(M1+M2)).f = (Mx2Tran M1).f + (Mx2Tran M2).f
proof
  set T12=Mx2Tran(M1+M2);
  set T2=Mx2Tran M2;
  set T1=Mx2Tran M1;
  per cases;
  suppose A1: n<>0;
    set L=LineVec2Mx@f;
    len f=n by CARD_1:def 7;
    then A4: width L=n by MATRIX13:1;
    A5: len M2=n & width M2=m by A1,MATRIX13:1;
    then A6: width(L*M2)=m by A4,MATRIX_3:def 4;
    A8: len M1=n by A1,MATRIX13:1;
    A9: width M1=m by A1,MATRIX13:1;
    then A10: width(L*M1)=m by A4,A8,MATRIX_3:def 4;
    len L=1 by MATRIX13:1;
    then len(L*M1)=1 by A4,A8,MATRIX_3:def 4; then
    B1: 1 in dom (L*M1) by FINSEQ_3:25;
    @(T1.f)=Line(L*M1,1) & @(T2.f)=Line(L*M2,1) by A1,Def3;
    hence T1.f+T2.f=Line(L*M1,1)+Line(L*M2,1) by RVSUM_1:def 4
     .=Line(L*M1+L*M2,1) by B1,A10,A6,MATRIX_4:59
     .=Line(L*(M1+M2),1) by A4,A8,A9,A5,MATRIX_4:62
     .=T12.f by A1,Def3;
  end;
  suppose A13: n=0;
    A14: 0.TOP-REAL m = 0* m by EUCLID:70 .= m |-> 0;
    then A15: T2.f = m |-> 0 by A13,Def3;
    thus T12.f = m |-> (zz+zz) by A13,A14,Def3
     .= m |-> zz + m |-> zz by RVSUM_1:14
     .= T1.f + T2.f by A13,A14,A15,Def3;
  end;
end;
