let n be Nat; :: thesis: for V being RealLinearSpace

for Ka being non void affinely-independent SimplicialComplex of V st |.Ka.| c= [#] Ka holds

degree Ka = degree (BCS (n,Ka))

let V be RealLinearSpace; :: thesis: for Ka being non void affinely-independent SimplicialComplex of V st |.Ka.| c= [#] Ka holds

degree Ka = degree (BCS (n,Ka))

let Ka be non void affinely-independent SimplicialComplex of V; :: thesis: ( |.Ka.| c= [#] Ka implies degree Ka = degree (BCS (n,Ka)) )

defpred S_{1}[ Nat] means ( degree Ka = degree (BCS ($1,Ka)) & not BCS ($1,Ka) is void & BCS ($1,Ka) is affinely-independent );

assume A1: |.Ka.| c= [#] Ka ; :: thesis: degree Ka = degree (BCS (n,Ka))

A2: for n being Nat st S_{1}[n] holds

S_{1}[n + 1]
_{1}[ 0 ]
by A1, Th16;

for n being Nat holds S_{1}[n]
from NAT_1:sch 2(A5, A2);

hence degree Ka = degree (BCS (n,Ka)) ; :: thesis: verum

for Ka being non void affinely-independent SimplicialComplex of V st |.Ka.| c= [#] Ka holds

degree Ka = degree (BCS (n,Ka))

let V be RealLinearSpace; :: thesis: for Ka being non void affinely-independent SimplicialComplex of V st |.Ka.| c= [#] Ka holds

degree Ka = degree (BCS (n,Ka))

let Ka be non void affinely-independent SimplicialComplex of V; :: thesis: ( |.Ka.| c= [#] Ka implies degree Ka = degree (BCS (n,Ka)) )

defpred S

assume A1: |.Ka.| c= [#] Ka ; :: thesis: degree Ka = degree (BCS (n,Ka))

A2: for n being Nat st S

S

proof

A5:
S
let n be Nat; :: thesis: ( S_{1}[n] implies S_{1}[n + 1] )

assume A3: S_{1}[n]
; :: thesis: S_{1}[n + 1]

A4: [#] (BCS (n,Ka)) = [#] Ka by A1, Th18;

( BCS ((n + 1),Ka) = BCS (BCS (n,Ka)) & |.(BCS (n,Ka)).| = |.Ka.| ) by A1, Th10, Th20;

hence S_{1}[n + 1]
by A1, A3, A4, Th28, Th31; :: thesis: verum

end;assume A3: S

A4: [#] (BCS (n,Ka)) = [#] Ka by A1, Th18;

( BCS ((n + 1),Ka) = BCS (BCS (n,Ka)) & |.(BCS (n,Ka)).| = |.Ka.| ) by A1, Th10, Th20;

hence S

for n being Nat holds S

hence degree Ka = degree (BCS (n,Ka)) ; :: thesis: verum