let V be RealLinearSpace; :: thesis: for K being non void total affinely-independent simplex-join-closed SimplicialComplex of V
for Aff being finite affinely-independent Subset of V
for B being Subset of V st Aff is Simplex of K holds
( B is Simplex of (BCS (Complex_of {Aff})) iff ( B is Simplex of (BCS K) & conv B c= conv Aff ) )

let K be non void total affinely-independent simplex-join-closed SimplicialComplex of V; :: thesis: for Aff being finite affinely-independent Subset of V
for B being Subset of V st Aff is Simplex of K holds
( B is Simplex of (BCS (Complex_of {Aff})) iff ( B is Simplex of (BCS K) & conv B c= conv Aff ) )

let Aff be finite affinely-independent Subset of V; :: thesis: for B being Subset of V st Aff is Simplex of K holds
( B is Simplex of (BCS (Complex_of {Aff})) iff ( B is Simplex of (BCS K) & conv B c= conv Aff ) )

let B be Subset of V; :: thesis: ( Aff is Simplex of K implies ( B is Simplex of (BCS (Complex_of {Aff})) iff ( B is Simplex of (BCS K) & conv B c= conv Aff ) ) )
set Bag = center_of_mass V;
set C = Complex_of {Aff};
A1: the topology of (Complex_of {Aff}) = bool Aff by SIMPLEX0:4;
assume Aff is Simplex of K ; :: thesis: ( B is Simplex of (BCS (Complex_of {Aff})) iff ( B is Simplex of (BCS K) & conv B c= conv Aff ) )
then reconsider s = Aff as Simplex of K ;
A2: [#] K = the carrier of V by SIMPLEX0:def 10;
then |.K.| c= [#] K ;
then A3: subdivision ((),K) = BCS K by Def5;
@ s is affinely-independent ;
then A4: Complex_of {Aff} is SubSimplicialComplex of K by Th3;
then the topology of (Complex_of {Aff}) c= the topology of K by SIMPLEX0:def 13;
then A5: |.(Complex_of {Aff}).| c= |.K.| by Th4;
[#] (Complex_of {Aff}) = [#] V ;
then A6: subdivision ((),(Complex_of {Aff})) = BCS (Complex_of {Aff}) by ;
then BCS (Complex_of {Aff}) is SubSimplicialComplex of BCS K by ;
then A7: the topology of (BCS (Complex_of {Aff})) c= the topology of (BCS K) by SIMPLEX0:def 13;
hereby :: thesis: ( B is Simplex of (BCS K) & conv B c= conv Aff implies B is Simplex of (BCS (Complex_of {Aff})) )
assume B is Simplex of (BCS (Complex_of {Aff})) ; :: thesis: ( B is Simplex of (BCS K) & conv B c= conv Aff )
then reconsider A = B as Simplex of (BCS (Complex_of {Aff})) ;
A in the topology of (BCS (Complex_of {Aff})) by PRE_TOPC:def 2;
then A in the topology of (BCS K) by A7;
then reconsider a = A as Simplex of (BCS K) by PRE_TOPC:def 2;
( |.(BCS (Complex_of {Aff})).| = |.(Complex_of {Aff}).| & conv (@ A) c= |.(BCS (Complex_of {Aff})).| ) by ;
then conv (@ a) c= conv Aff by Th8;
hence ( B is Simplex of (BCS K) & conv B c= conv Aff ) ; :: thesis: verum
end;
assume that
A8: B is Simplex of (BCS K) and
A9: conv B c= conv Aff ; :: thesis: B is Simplex of (BCS (Complex_of {Aff}))
reconsider A = B as Simplex of (BCS K) by A8;
consider SS being c=-linear finite simplex-like Subset-Family of K such that
A10: B = () .: SS by ;
reconsider ss = SS as c=-linear finite Subset-Family of (Complex_of {Aff}) by A2;
[#] (subdivision ((),(Complex_of {Aff}))) = [#] (Complex_of {Aff}) by SIMPLEX0:def 20;
then reconsider Bss = () .: ss as Subset of (BCS (Complex_of {Aff})) by ;
A11: dom () = (bool the carrier of V) \ by FUNCT_2:def 1;
ss is simplex-like
proof
let a be Subset of (Complex_of {Aff}); :: according to TOPS_2:def 1 :: thesis: ( not a in ss or not a is dependent )
assume A12: a in ss ; :: thesis: not a is dependent
reconsider aK = a as Simplex of K by ;
per cases ( aK is empty or not aK is empty ) ;
end;
end;
then Bss is simplex-like by ;
hence B is Simplex of (BCS (Complex_of {Aff})) by A10; :: thesis: verum