reserve
  a for natural Number,
  k,l,m,n,k1,b,c,i for Nat,
  x,y,z,y1,y2 for object,
  X,Y for set,
  f,g for Function;
reserve p,q,r,s,t for FinSequence;

theorem Th17:
  a <= len p & q = p|(Seg a) implies len q = a & dom q = Seg a
proof
  assume that
A1: a <= len p and
A2: q = p|(Seg a);
  Seg a c= Seg len p by A1,Th5;
  then Seg a c= dom p by Def3;
  then a in NAT & dom q = Seg a by A2,ORDINAL1:def 12,RELAT_1:62;
  hence thesis by Def3;
end;
