reserve i,j,n,k,m for Nat,
     a,b,x,y,z for object,
     F,G for FinSequence-yielding FinSequence,
     f,g,p,q for FinSequence,
     X,Y for set,
     D for non empty set;
reserve
  B,A,M for BinOp of D,
  F,G for D* -valued FinSequence,
  f for FinSequence of D,
  d,d1,d2 for Element of D;
reserve
  F,G for non-empty non empty FinSequence of D*,
  f for non empty FinSequence of D;
reserve f,g for FinSequence of D,
        a,b,c for set,
        F,F1,F2 for finite set;

theorem Th122:
  for f,g be FinSequence holds
    f in doms(k,n) & g in doms(k,m) implies f^g in doms(k,n+m)
proof
  let f,g be FinSequence;
  assume
A1: f in doms(k,n) & g in doms(k,m);
  then consider p1 be Element of (Seg k)* such that
A2: p1=f & len p1= n;
  consider s1 be Element of (Seg k)* such that
A3: s1=g & len s1= m by A1;
A4: len (f^g) = n+m by A2,A3,FINSEQ_1:22;
A5: rng (f^g) = rng f \/ rng g by FINSEQ_1:31;
  rng s1 c= Seg k & rng p1 c= Seg k;
  then f^g is FinSequence of (Seg k) by FINSEQ_1:def 4,A5,A2,A3,XBOOLE_1:8;
  then f^g is Element of (Seg k)* by FINSEQ_1:def 11;
  hence thesis by A4;
end;
