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
  p just_once_values x iff x in rng p & rng(p |-- x) misses {x}
proof
  thus p just_once_values x implies x in rng p & rng(p |-- x) misses {x}
  proof
    assume
A1: p just_once_values x;
    hence x in rng p by Th45;
    assume not rng(p |-- x) misses {x};
    then
A2: ex y being object st y in rng(p |-- x) & y in {x} by XBOOLE_0:3;
    not x in rng(p |-- x) by A1,Th45;
    hence thesis by A2,TARSKI:def 1;
  end;
  assume that
A3: x in rng p and
A4: rng(p |-- x) misses {x};
  now
A5: x in {x} by TARSKI:def 1;
    assume x in rng(p |-- x);
    hence contradiction by A4,A5,XBOOLE_0:3;
  end;
  hence thesis by A3,Th45;
end;
