 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
  for A be Matrix of n,m,F_Real,
      B be Matrix of k,m,F_Real
  for g be k-element real-valued FinSequence holds
    (Mx2Tran(A^B)).(f^g) = (Mx2Tran A).f+(Mx2Tran B).g
proof
  A1: len f=n by CARD_1:def 7;
  rng f c=REAL;
  then f is FinSequence of REAL by FINSEQ_1:def 4;
  then reconsider F=f,n0=n|->In(0,REAL)
as Element of n-tuples_on REAL by A1,FINSEQ_2:92;
  let A be Matrix of n,m,F_Real,B be Matrix of k,m,F_Real;
  let g be k-element real-valued FinSequence;
  A2: len g=k by CARD_1:def 7;
  rng g c=REAL;
  then g is FinSequence of REAL by FINSEQ_1:def 4;
  then reconsider G=g,k0=k|->In(0,REAL) as Element of k-tuples_on REAL
    by A2,FINSEQ_2:92;
  f=F+n0 by RVSUM_1:16;
  then g=G+k0 & f=addreal.:(f,n0) by RVSUM_1:16,def 4;
  then f^g=(addreal.:(F,n0))^(addreal.:(k0,G)) by RVSUM_1:def 4
   .=addreal.:(F^k0,n0^G) by FINSEQOP:11
   .=(F^k0)+(n0^G) by RVSUM_1:def 4;
  hence (Mx2Tran(A^B)).(f^g)=(Mx2Tran(A^B)).(F^k0)+(Mx2Tran(A^B)).(n0^G)
  by Th22
   .=(Mx2Tran A).f+(Mx2Tran(A^B)).(n0^G) by Th35
   .=(Mx2Tran A).f+(Mx2Tran B).g by Th35;
end;
