let V be RealLinearSpace; :: thesis: for K being non void total affinely-independent simplex-join-closed SimplicialComplex of V
for Sk being finite simplex-like Subset-Family of K
for Ak being Simplex of K st Sk is c=-linear & Sk is with_non-empty_elements & card Sk = card (union Sk) & union Sk c= Ak & card Ak = (card Sk) + 1 holds
{ S1 where S1 is Simplex of card Sk, BCS K : ( () .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {((() .: Sk) \/ (() .: {Ak}))}

let K be non void total affinely-independent simplex-join-closed SimplicialComplex of V; :: thesis: for Sk being finite simplex-like Subset-Family of K
for Ak being Simplex of K st Sk is c=-linear & Sk is with_non-empty_elements & card Sk = card (union Sk) & union Sk c= Ak & card Ak = (card Sk) + 1 holds
{ S1 where S1 is Simplex of card Sk, BCS K : ( () .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {((() .: Sk) \/ (() .: {Ak}))}

let Sk be finite simplex-like Subset-Family of K; :: thesis: for Ak being Simplex of K st Sk is c=-linear & Sk is with_non-empty_elements & card Sk = card (union Sk) & union Sk c= Ak & card Ak = (card Sk) + 1 holds
{ S1 where S1 is Simplex of card Sk, BCS K : ( () .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {((() .: Sk) \/ (() .: {Ak}))}

let Ak be Simplex of K; :: thesis: ( Sk is c=-linear & Sk is with_non-empty_elements & card Sk = card (union Sk) & union Sk c= Ak & card Ak = (card Sk) + 1 implies { S1 where S1 is Simplex of card Sk, BCS K : ( () .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {((() .: Sk) \/ (() .: {Ak}))} )
set B = center_of_mass V;
assume that
A1: ( Sk is c=-linear & Sk is with_non-empty_elements ) and
A2: card Sk = card (union Sk) and
A3: union Sk c= Ak and
A4: card Ak = (card Sk) + 1 ; :: thesis: { S1 where S1 is Simplex of card Sk, BCS K : ( () .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {((() .: Sk) \/ (() .: {Ak}))}
card (Ak \ (union Sk)) = ((card Sk) + 1) - (card Sk) by
.= 1 ;
then consider v being object such that
A5: Ak \ (union Sk) = {v} by CARD_2:42;
reconsider Ak1 = @ Ak as finite affinely-independent Subset of V ;
set C = Complex_of {Ak1};
reconsider c = card Ak as ExtReal ;
A6: degree (Complex_of {Ak1}) = c - 1 by SIMPLEX0:26
.= (card Ak) + (- 1) by XXREAL_3:def 2
.= card Sk by A4 ;
reconsider Sk1 = @ Sk as c=-linear finite finite-membered Subset-Family of V by A1;
set XX = { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : () .: Sk c= W } ;
set YY = { W where W is Simplex of card Sk, BCS K : ( () .: Sk c= W & conv (@ W) c= conv (@ Ak) ) } ;
[#] K = the carrier of V by SIMPLEX0:def 10;
then |.K.| c= [#] K ;
then A7: subdivision ((),K) = BCS K by Def5;
A8: Complex_of {Ak1} is SubSimplicialComplex of K by Th3;
then the topology of (Complex_of {Ak1}) c= the topology of K by SIMPLEX0:def 13;
then A9: |.(Complex_of {Ak1}).| c= |.K.| by Th4;
A10: [#] (Complex_of {Ak1}) = [#] V ;
then A11: degree (Complex_of {Ak1}) = degree (BCS (Complex_of {Ak1})) by ;
subdivision ((),(Complex_of {Ak1})) = BCS (Complex_of {Ak1}) by ;
then BCS (Complex_of {Ak1}) is SubSimplicialComplex of BCS K by ;
then A12: degree (BCS (Complex_of {Ak1})) <= degree (BCS K) by SIMPLEX0:32;
A13: { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : () .: Sk c= W } c= { W where W is Simplex of card Sk, BCS K : ( () .: Sk c= W & conv (@ W) c= conv (@ Ak) ) }
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : () .: Sk c= W } or x in { W where W is Simplex of card Sk, BCS K : ( () .: Sk c= W & conv (@ W) c= conv (@ Ak) ) } )
assume x in { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : () .: Sk c= W } ; :: thesis: x in { W where W is Simplex of card Sk, BCS K : ( () .: Sk c= W & conv (@ W) c= conv (@ Ak) ) }
then consider W being Simplex of card Sk, BCS (Complex_of {Ak1}) such that
A14: ( x = W & () .: Sk1 c= W ) ;
W = @ W ;
then reconsider w = W as Simplex of (BCS K) by Th40;
card W = (degree (BCS (Complex_of {Ak1}))) + 1 by ;
then A15: w is Simplex of card Sk, BCS K by ;
( conv (@ W) c= conv (@ Ak) & @ w = w ) by Th40;
hence x in { W where W is Simplex of card Sk, BCS K : ( () .: Sk c= W & conv (@ W) c= conv (@ Ak) ) } by ; :: thesis: verum
end;
A16: [#] (subdivision ((),(Complex_of {Ak1}))) = [#] (Complex_of {Ak1}) by SIMPLEX0:def 20;
A17: { W where W is Simplex of card Sk, BCS K : ( () .: Sk c= W & conv (@ W) c= conv (@ Ak) ) } c= { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : () .: Sk c= W }
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { W where W is Simplex of card Sk, BCS K : ( () .: Sk c= W & conv (@ W) c= conv (@ Ak) ) } or x in { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : () .: Sk c= W } )
reconsider c1 = card Sk as ExtReal ;
assume x in { W where W is Simplex of card Sk, BCS K : ( () .: Sk c= W & conv (@ W) c= conv (@ Ak) ) } ; :: thesis: x in { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : () .: Sk c= W }
then consider W being Simplex of card Sk, BCS K such that
A18: ( W = x & () .: Sk c= W ) and
A19: conv (@ W) c= conv (@ Ak) ;
reconsider w = @ W as Subset of (BCS (Complex_of {Ak1})) by ;
reconsider cW = card W as ExtReal ;
card W = c1 + 1 by
.= (card Sk) + 1 by XXREAL_3:def 2 ;
then card Sk = (card W) + (- 1) ;
then A20: card Sk = cW - 1 by XXREAL_3:def 2;
w is simplex-like by ;
then w is Simplex of card Sk, BCS (Complex_of {Ak1}) by ;
hence x in { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : () .: Sk c= W } by A18; :: thesis: verum
end;
v in {v} by TARSKI:def 1;
then A21: ( v in Ak1 & not v in union Sk ) by ;
Ak = Ak \/ (union Sk) by
.= {v} \/ (union Sk1) by ;
then { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : () .: Sk c= W } = {((() .: Sk1) \/ (() .: {Ak}))} by A1, A2, A21, Th38;
hence { S1 where S1 is Simplex of card Sk, BCS K : ( () .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {((() .: Sk) \/ (() .: {Ak}))} by ; :: thesis: verum