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 : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {(((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: {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 : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {(((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: {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 : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {(((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: {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 : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {(((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: {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 : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {(((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: {Ak}))}

card (Ak \ (union Sk)) = ((card Sk) + 1) - (card Sk) by A2, A3, A4, CARD_2:44

.= 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}) : (center_of_mass V) .: Sk c= W } ;

set YY = { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ Ak) ) } ;

[#] K = the carrier of V by SIMPLEX0:def 10;

then |.K.| c= [#] K ;

then A7: subdivision ((center_of_mass V),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 A9, Th31;

subdivision ((center_of_mass V),(Complex_of {Ak1})) = BCS (Complex_of {Ak1}) by A9, A10, Def5;

then BCS (Complex_of {Ak1}) is SubSimplicialComplex of BCS K by A7, A8, SIMPLEX0:58;

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}) : (center_of_mass V) .: Sk c= W } c= { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ Ak) ) }

A17: { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ Ak) ) } c= { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : (center_of_mass V) .: Sk c= W }

then A21: ( v in Ak1 & not v in union Sk ) by A5, XBOOLE_0:def 5;

Ak = Ak \/ (union Sk) by A3, XBOOLE_1:12

.= {v} \/ (union Sk1) by A5, XBOOLE_1:39 ;

then { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : (center_of_mass V) .: Sk c= W } = {(((center_of_mass V) .: Sk1) \/ ((center_of_mass V) .: {Ak}))} by A1, A2, A21, Th38;

hence { S1 where S1 is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {(((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: {Ak}))} by A13, A17; :: thesis: verum

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 : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {(((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: {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 : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {(((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: {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 : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {(((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: {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 : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {(((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: {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 : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {(((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: {Ak}))}

card (Ak \ (union Sk)) = ((card Sk) + 1) - (card Sk) by A2, A3, A4, CARD_2:44

.= 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}) : (center_of_mass V) .: Sk c= W } ;

set YY = { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ Ak) ) } ;

[#] K = the carrier of V by SIMPLEX0:def 10;

then |.K.| c= [#] K ;

then A7: subdivision ((center_of_mass V),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 A9, Th31;

subdivision ((center_of_mass V),(Complex_of {Ak1})) = BCS (Complex_of {Ak1}) by A9, A10, Def5;

then BCS (Complex_of {Ak1}) is SubSimplicialComplex of BCS K by A7, A8, SIMPLEX0:58;

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}) : (center_of_mass V) .: Sk c= W } c= { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ Ak) ) }

proof

A16:
[#] (subdivision ((center_of_mass V),(Complex_of {Ak1}))) = [#] (Complex_of {Ak1})
by SIMPLEX0:def 20;
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}) : (center_of_mass V) .: Sk c= W } or x in { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ Ak) ) } )

assume x in { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : (center_of_mass V) .: Sk c= W } ; :: thesis: x in { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: 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 & (center_of_mass V) .: 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 A6, A11, SIMPLEX0:def 18;

then A15: w is Simplex of card Sk, BCS K by A6, A11, A12, SIMPLEX0:def 18;

( conv (@ W) c= conv (@ Ak) & @ w = w ) by Th40;

hence x in { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ Ak) ) } by A14, A15; :: thesis: verum

end;assume x in { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : (center_of_mass V) .: Sk c= W } ; :: thesis: x in { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: 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 & (center_of_mass V) .: 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 A6, A11, SIMPLEX0:def 18;

then A15: w is Simplex of card Sk, BCS K by A6, A11, A12, SIMPLEX0:def 18;

( conv (@ W) c= conv (@ Ak) & @ w = w ) by Th40;

hence x in { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ Ak) ) } by A14, A15; :: thesis: verum

A17: { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ Ak) ) } c= { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : (center_of_mass V) .: Sk c= W }

proof

v in {v}
by TARSKI:def 1;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ Ak) ) } or x in { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : (center_of_mass V) .: Sk c= W } )

reconsider c1 = card Sk as ExtReal ;

assume x in { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ Ak) ) } ; :: thesis: x in { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : (center_of_mass V) .: Sk c= W }

then consider W being Simplex of card Sk, BCS K such that

A18: ( W = x & (center_of_mass V) .: Sk c= W ) and

A19: conv (@ W) c= conv (@ Ak) ;

reconsider w = @ W as Subset of (BCS (Complex_of {Ak1})) by A9, A16, Def5;

reconsider cW = card W as ExtReal ;

card W = c1 + 1 by A6, A11, A12, SIMPLEX0:def 18

.= (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 A19, Th40;

then w is Simplex of card Sk, BCS (Complex_of {Ak1}) by A20, SIMPLEX0:48;

hence x in { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : (center_of_mass V) .: Sk c= W } by A18; :: thesis: verum

end;reconsider c1 = card Sk as ExtReal ;

assume x in { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ Ak) ) } ; :: thesis: x in { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : (center_of_mass V) .: Sk c= W }

then consider W being Simplex of card Sk, BCS K such that

A18: ( W = x & (center_of_mass V) .: Sk c= W ) and

A19: conv (@ W) c= conv (@ Ak) ;

reconsider w = @ W as Subset of (BCS (Complex_of {Ak1})) by A9, A16, Def5;

reconsider cW = card W as ExtReal ;

card W = c1 + 1 by A6, A11, A12, SIMPLEX0:def 18

.= (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 A19, Th40;

then w is Simplex of card Sk, BCS (Complex_of {Ak1}) by A20, SIMPLEX0:48;

hence x in { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : (center_of_mass V) .: Sk c= W } by A18; :: thesis: verum

then A21: ( v in Ak1 & not v in union Sk ) by A5, XBOOLE_0:def 5;

Ak = Ak \/ (union Sk) by A3, XBOOLE_1:12

.= {v} \/ (union Sk1) by A5, XBOOLE_1:39 ;

then { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : (center_of_mass V) .: Sk c= W } = {(((center_of_mass V) .: Sk1) \/ ((center_of_mass V) .: {Ak}))} by A1, A2, A21, Th38;

hence { S1 where S1 is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {(((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: {Ak}))} by A13, A17; :: thesis: verum