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 Th122:
  ((a,b) followed_by c).1 = b
proof
  dom (0,1) --> (a,b) = {0,1} by FUNCT_4:62;
  then
A1: 1 in dom (0,1) --> (a,b) by TARSKI:def 2;
  thus ((a,b) followed_by c).1 = ((NAT --> c) +* ((0,1) --> (a,b))).1
    .= ((0,1) --> (a,b)).1 by A1,FUNCT_4:13
    .= b by FUNCT_4:63;
end;
