let n be Nat; 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; 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; ( |.Ka.| c= [#] Ka implies degree Ka = degree (BCS (n,Ka)) )
defpred S1[ 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
; degree Ka = degree (BCS (n,Ka))
A2:
for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be
Nat;
( S1[n] implies S1[n + 1] )
assume A3:
S1[
n]
;
S1[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
S1[
n + 1]
by A1, A3, A4, Th28, Th31;
verum
end;
A5:
S1[ 0 ]
by A1, Th16;
for n being Nat holds S1[n]
from NAT_1:sch 2(A5, A2);
hence
degree Ka = degree (BCS (n,Ka))
; verum