reserve x,y, X,Y,Z for set,
        D for non empty set,
        n,k for Nat,
        i,i1,i2 for Integer;
reserve K for SimplicialComplexStr;

theorem
  for A be finite Subset of X holds degree Complex_of {A} = card A - 1
 proof
  let A be finite Subset of X;
  set C=Complex_of{A};
  A1: the topology of C=bool A by Th4;
  then for S be finite Subset of C st S is simplex-like holds
   card S<=(card A-1)+1 by NAT_1:43;
  then A2: degree C<=card A-1 by Th25;
  A c=A;
  then reconsider A1=A as finite Simplex of C by A1,PRE_TOPC:def 2;
  card A=card A-1+1 & card A1<=degree C+1 by Def12;
  then card A-1<=degree C by XREAL_1:8;
  hence card A-1=degree C by A2,XXREAL_0:1;
 end;
