reserve i,j,k,n,m for Nat,
  x,y,z,y1,y2 for object, X,Y,D for set,
  p,q for XFinSequence;
reserve k1,k2 for Nat;
reserve D for non empty set,
  F,G for XFinSequence of D,
  b for BinOp of D,
  d,d1,d2 for Element of D;
reserve F for XFinSequence,
        rF,rF1,rF2 for real-valued XFinSequence,
        r for Real,
        cF,cF1,cF2 for complex-valued XFinSequence,
        c,c1,c2 for Complex;

theorem Th69:
  for f be Function,p,q,fp,fq be XFinSequence st
       rng p c= dom f & rng q c= dom f & fp = f*p & fq = f*q
    holds fp ^ fq = f*(p^q)
proof
  let f be Function,p,q,fp,fq be XFinSequence such that
A1:       rng p c= dom f & rng q c= dom f & fp = f*p & fq = f*q;
set pq=p^q;
A2:rng pq = rng p \/rng q by AFINSQ_1:26;
then A3:dom (f*pq)=dom pq by A1,RELAT_1:27,XBOOLE_1:8;
reconsider fpq = f*pq as XFinSequence by A2,A1,AFINSQ_1:10,XBOOLE_1:8;
A4:dom fp=dom p & dom fq = dom q by A1,RELAT_1:27;
A5:dom pq=len p+len q & dom (fp^fq) = len fp+len fq by AFINSQ_1:def 3;
A6:len fpq = len (fp^fq) by A2,A1,A4,A5,RELAT_1:27,XBOOLE_1:8;
k < len fpq implies (fp^fq).k = fpq.k
proof
  assume A7:k< len fpq;
  then A8:k in dom fpq by AFINSQ_1:86;
  per cases;
   suppose k < len p;
      then k in dom p by AFINSQ_1:86;
      then pq.k = p.k & fp.k = f.(p.k) & (fp^fq).k =fp.k
        by A1,A4,AFINSQ_1:def 3,FUNCT_1:13;
      hence thesis by A8,FUNCT_1:12;
   end;
   suppose A9:k >= len p;
      then reconsider kp=k-len p as Element of NAT by NAT_1:21;
      len p + kp < len p+len q by A5,A2,A1,A7,RELAT_1:27,XBOOLE_1:8;
      then
      kp < len q by XREAL_1:7;
      then pq.k = q.kp & (fp^fq).k = fq.kp & fq.kp = f.(q.kp)
          by A7,A1,A3,A4,A5,A9,AFINSQ_1:18,FUNCT_1:13,AFINSQ_1:86;
      hence thesis by A8,FUNCT_1:12;
   end;
end;
hence thesis by A6,AFINSQ_1:9;
end;
