reserve a,x,y for object, A,B for set,
  l,m,n for Nat;
reserve X,Y for set, x for object,
  p,q for Function-yielding FinSequence,
  f,g,h for Function;
reserve m,n,k for Nat, R for Relation;
reserve i,j for Nat;
reserve F for Function,
  e,x,y,z for object;
reserve a,b,c for set;

theorem Th125:
  rng(a followed_by b) = {a,b}
proof
  rng(NAT --> b) = {b} by FUNCOP_1:8;
  then rng((NAT --> b)+*(0,a)) c= {b} \/ {a} by Th99;
  hence rng(a followed_by b) c= {a,b} by ENUMSET1:1;
  let x be object;
  assume
A1: x in {a,b};
A2: dom(NAT --> b) = NAT;
  1 in dom(NAT --> b);
  then
A3: 1 in dom(a followed_by b) by Th29;
  (a followed_by b).1 = b by Th119;
  then
A4: b in rng(a followed_by b) by A3,FUNCT_1:3;
  a in rng(a followed_by b) by A2,Th101;
  hence thesis by A1,A4,TARSKI:def 2;
end;
