reserve x,y for object,X,Y for set,
  D for non empty set,
  i,j,k,l,m,n,m9,n9 for Nat,
  i0,j0,n0,m0 for non zero Nat,
  K for Field,
  a,b for Element of K,
  p for FinSequence of K,
  M for Matrix of n,K;
reserve A for (Matrix of D),
  A9 for Matrix of n9,m9,D,
  M9 for Matrix of n9, m9,K,
  nt,nt1,nt2 for Element of n-tuples_on NAT,
  mt,mt1 for Element of m -tuples_on NAT,
  M for Matrix of K;
reserve P,P1,P2,Q,Q1,Q2 for without_zero finite Subset of NAT;

theorem Th88:
  for p,pf,q,qf be FinSequence of K, f be Function st pf = p * f &
  rng f c= dom p & qf = q * f & rng f c= dom q holds (p+q) * f = pf + qf
proof
  let p,pf,q,qf be FinSequence of K, f be Function such that
A1: pf = p * f and
A2: rng f c= dom p and
A3: qf = q * f and
A4: rng f c= dom q;
A5: dom pf=dom f by A1,A2,RELAT_1:27;
  set KK=the carrier of K;
A6: dom pf=Seg len pf by FINSEQ_1:def 3;
A7: dom qf=dom f by A3,A4,RELAT_1:27;
  then len qf=len pf by A5,A6,FINSEQ_1:def 3;
  then reconsider pf9=pf,qf9=qf as Element of (len pf)-tuples_on KK by
FINSEQ_2:92;
A8: dom (pf9+qf9)=dom f by A5,A6,FINSEQ_2:124;
  set pq=p+q;
A9: rng q c= KK by FINSEQ_1:def 4;
  rng p c= KK by FINSEQ_1:def 4;
  then [:rng p,rng q:]c=[:KK,KK:] by A9,ZFMISC_1:96;
  then [:rng p, rng q:] c= dom the addF of K by FUNCT_2:def 1;
  then
A10: dom pq=dom p /\ dom q by FUNCOP_1:69;
  then
A11: rng f c= dom pq by A2,A4,XBOOLE_1:19;
A12: now
A13: rng qf c= KK by FINSEQ_1:def 4;
A14: rng pf c= KK by FINSEQ_1:def 4;
    let x being object such that
A15: x in dom f;
A16: f.x in rng f by A15,FUNCT_1:def 3;
    dom p=Seg len p by FINSEQ_1:def 3;
    then f.x in Seg len p by A2,A16;
    then reconsider n=x,fx=f.x as Element of NAT by A5,A15;
A17: qf.x in rng qf by A7,A15,FUNCT_1:def 3;
    pf.x in rng pf by A5,A15,FUNCT_1:def 3;
    then reconsider pfn=pf.n,qfn=qf.n as Element of K by A14,A17,A13;
A18: pfn=p.fx by A1,A15,FUNCT_1:13;
A19: qfn=q.fx by A3,A15,FUNCT_1:13;
    thus (pq*f).x = pq.fx by A15,FUNCT_1:13
      .= pfn+qfn by A11,A16,A18,A19,FVSUM_1:17
      .= (pf9+qf9).x by A5,A6,A15,FVSUM_1:18;
  end;
  dom (pq*f)=dom f by A2,A4,A10,RELAT_1:27,XBOOLE_1:19;
  hence thesis by A8,A12;
end;
