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;

theorem
  p just_once_values x iff x in rng p & {x..p} = p " {x}
proof
  thus p just_once_values x implies x in rng p & {x..p} = p " {x}
  proof
    assume
A1: p just_once_values x;
    then x..p = p <- x by Th25;
    hence thesis by A1,Th5,Th11;
  end;
  assume that
A2: x in rng p and
A3: {x..p} = p " {x};
A4: now
    let z be object;
    assume that
A5: z in dom p and
A6: z <> x..p and
A7: p.z = x;
    p.z in {x} by A7,TARSKI:def 1;
    then z in p " {x} by A5,FUNCT_1:def 7;
    hence contradiction by A3,A6,TARSKI:def 1;
  end;
  p.(x..p) = x & x..p in dom p by A2,Th19,Th20;
  hence thesis by A4,Th7;
end;
