 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 Th22:
  (Mx2Tran M).(f1+f2) = (Mx2Tran M).f1 + (Mx2Tran M).f2
proof
  set f12=f1+f2;
  set T=Mx2Tran M;
  per cases;
  suppose A1: n<>0;
    A4: len M=n by A1,MATRIX13:1;
    set L2=LineVec2Mx@f2;
    set L1=LineVec2Mx@f1;
    A5: len L2=1 by MATRIX13:1;
    A6: len f2=n by CARD_1:def 7;
    then A7: width L2=n by MATRIX13:1;
    A8: width M=m by A1,MATRIX13:1;
    then A9: width(L2*M)=m by A7,A4,MATRIX_3:def 4;
    A10: len f1=n by CARD_1:def 7;
    then A11: width L1=n by MATRIX13:1;
    then A12: width(L1*M)=m by A4,A8,MATRIX_3:def 4;
    A13: len L1=1 by MATRIX13:1;
    then a13: len(L1*M)=1 by A11,A4,MATRIX_3:def 4;
    A15: @(T.f1)=Line(L1*M,1) & @(T.f2)=Line(L2*M,1) by A1,Def3;
    B1: 1 in dom (L1*M) by a13,FINSEQ_3:25;
    @f12=@f1+@f2 by RVSUM_1:def 4;
    then (LineVec2Mx@f12)*M=(L1+L2)*M by A10,A6,MATRIX15:27
     .=L1*M+L2*M by A13,A5,A11,A7,A4,MATRIX_4:63;
    hence T.f12=Line(L1*M+L2*M,1) by A1,Def3
     .=Line(L1*M,1)+Line(L2*M,1) by B1,A12,A9,MATRIX_4:59
     .=T.f1+T.f2 by A15,RVSUM_1:def 4;
  end;
  suppose A16: n=0;
reconsider zz=0 as Real;
    A17: 0.TOP-REAL m = 0* m by EUCLID:70 .= m |-> 0;
    then A18: T.f2 = m |-> 0 by A16,Def3;
    thus T.(f1+f2) = m |-> (zz+zz) by A16,A17,Def3
     .= m |-> zz + m |-> zz by RVSUM_1:14
     .= T.f1 + T.f2 by A16,A18;
  end;
end;
