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 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 ; :: thesis: degree Ka = degree (BCS n,Ka)
A2: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A3: S₁[n] ; :: thesis: 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; :: thesis: 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) ; :: thesis: verum