reserve f for Function;
reserve p,q for FinSequence;
reserve A,B,C for set,x,x1,x2,y,z for object;
reserve k,l,m,n for Nat;
reserve a for Nat;
reserve D for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve L,M for Element of NAT;

theorem
  x in rng p & p is one-to-one implies p - {x} = (p -| x) ^ (p |-- x)
proof
  assume x in rng p & p is one-to-one;
  then p just_once_values x by Th8;
  hence thesis by Th54;
end;
