let T be Tree; :: thesis: for P being AntiChain_of_Prefixes of T holds { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_proper_prefix_of t1 } \ { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t1 } = P
let P be AntiChain_of_Prefixes of T; :: thesis: { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_proper_prefix_of t1 } \ { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t1 } = P

now :: thesis: for x being object st x in { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_proper_prefix_of t1 } \ { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t1 } holds
x in P

let x be object ; :: thesis: ( x in { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_proper_prefix_of t1 } \ { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t1 } implies x in P )
assume A1: x in { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_proper_prefix_of t1 } \ { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t1 } ; :: thesis: x in P
then A2: x in { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_proper_prefix_of t1 } ;
A3: not x in { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t1 } by A1, XBOOLE_0:def 5;
assume A4: not x in P ; :: thesis: contradiction
ex t1 being Element of T st
( x = t1 & ( for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t1 ) )

proof

consider t9 being Element of T such that
A5: x = t9 and
A6: for p being FinSequence of NAT st p in P holds
not p is_a_proper_prefix_of t9 by A2;

now :: thesis: for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t9

let p be FinSequence of NAT ; :: thesis: ( p in P implies not b₁ is_a_prefix_of t9 )
assume A7: p in P ; :: thesis: not b₁ is_a_prefix_of t9
then A8: not p is_a_proper_prefix_of t9 by A6;

per cases ( not p is_a_prefix_of t9 or p = t9 ) by A8;

suppose not p is_a_prefix_of t9 ; :: thesis: not b₁ is_a_prefix_of t9

hence not p is_a_prefix_of t9 ; :: thesis: verum

end;

suppose p = t9 ; :: thesis: not b₁ is_a_prefix_of t9

hence not p is_a_prefix_of t9 by A4, A5, A7; :: thesis: verum

end;

end;

end;

hence ex t1 being Element of T st
( x = t1 & ( for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t1 ) ) by A5; :: thesis: verum

end;

hence contradiction by A3; :: thesis: verum

end;

hence { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_proper_prefix_of t1 } \ { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t1 } c= P ; :: according to XBOOLE_0:def 10 :: thesis: P c= { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_proper_prefix_of t1 } \ { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t1 }
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in P or x in { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_proper_prefix_of t1 } \ { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t1 } )
assume A9: x in P ; :: thesis: x in { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_proper_prefix_of t1 } \ { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t1 }
A10: P c= { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_proper_prefix_of t1 } by Th4;
not x in { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t1 }

proof

assume x in { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t1 } ; :: thesis: contradiction
then ex t9 being Element of T st
( x = t9 & ( for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t9 ) ) ;
hence contradiction by A9; :: thesis: verum

end;

hence x in { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_proper_prefix_of t1 } \ { t1 where t1 is Element of T : for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t1 } by A9, A10, XBOOLE_0:def 5; :: thesis: verum