set n = {} ;
let L be non empty doubleLoopStr ; :: thesis: for p being Polynomial of {},L ex a being Element of L st p = {(EmptyBag {})} --> a
let p be Polynomial of {},L; :: thesis: ex a being Element of L st p = {(EmptyBag {})} --> a
A1: for b being bag of {} holds b = {}

proof

let b be bag of {} ; :: thesis: b = {}
b in Bags {} by PRE_POLY:def 12;
hence b = {} by PRE_POLY:51, TARSKI:def 1; :: thesis: verum

end;

reconsider p = p as Function of (Bags {}),L ;
reconsider p = p as Function of {{}}, the carrier of L by PRE_POLY:51;
set a = p /. {};
A2: dom p = {{}} by FUNCT_2:def 1
.= {(EmptyBag {})} by A1 ;
A3: for u being set st u in p holds
u in [:{(EmptyBag {})},{(p /. {})}:]

proof

let u be set ; :: thesis: ( u in p implies u in [:{(EmptyBag {})},{(p /. {})}:] )
assume A4: u in p ; :: thesis: u in [:{(EmptyBag {})},{(p /. {})}:]
then consider p1, p2 being set such that
A5: u = [p1,p2] by RELAT_1:def 1;
A6: p1 in dom p by A4, A5, RELAT_1:def 4;
then reconsider p1 = p1 as bag of {} by A2;
A7: p1 = {} by A1;
then p2 = p . {} by A4, A5, A6, FUNCT_1:def 2
.= p /. {} by A6, A7, PARTFUN1:def 6 ;
then p2 in {(p /. {})} by TARSKI:def 1;
hence u in [:{(EmptyBag {})},{(p /. {})}:] by A2, A5, A6, ZFMISC_1:def 2; :: thesis: verum

end;

take p /. {} ; :: thesis: p = {(EmptyBag {})} --> (p /. {})
A8: EmptyBag {} = {} by A1;
for u being set st u in [:{(EmptyBag {})},{(p /. {})}:] holds
u in p

proof

let u be set ; :: thesis: ( u in [:{(EmptyBag {})},{(p /. {})}:] implies u in p )
assume u in [:{(EmptyBag {})},{(p /. {})}:] ; :: thesis: u in p
then consider u1, u2 being set such that
A9: u1 in {(EmptyBag {})} and
A10: u2 in {(p /. {})} and
A11: u = [u1,u2] by ZFMISC_1:def 2;
A12: u1 = {} by A8, A9, TARSKI:def 1;
u2 = p /. {} by A10, TARSKI:def 1
.= p . {} by A2, A9, A12, PARTFUN1:def 6 ;
hence u in p by A2, A9, A11, A12, FUNCT_1:1; :: thesis: verum

end;

hence p = {(EmptyBag {})} --> (p /. {}) by A3, TARSKI:1; :: thesis: verum