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;
reserve KX for SimplicialComplexStr of X,
        SX for SubSimplicialComplex of KX;
reserve SC for SimplicialComplex of X;
reserve K for non void subset-closed SimplicialComplexStr;

theorem
  for S be Simplex of n,K st n<=degree K holds
    X is Face of S iff ex x st x in S & S\{x} = X
 proof
  let S be Simplex of n,K such that
   A1: n<=degree K;
  n-1<=n-0 by XREAL_1:6;
  then A2: n-1<=degree K by A1,XXREAL_0:2;
  reconsider N=n as Integer;
  A3: n-1>=0-1 by XREAL_1:6;
  then A4: max(n-1,-1)=n-1 by XXREAL_0:def 10;
  A5: card S=N+1 by A1,Def18;
  hereby assume X is Face of S;
   then reconsider f=X as Face of S;
   A6: f c=S by A1,Def19;
   card f=n-1+1 by A2,A3,A4,Def18;
   then card(S\f)=n+1-n by A5,A6,CARD_2:44
    .=1;
   then consider x being object such that
    A7: S\f={x} by CARD_2:42;
    reconsider x as set by TARSKI:1;
   take x;
   x in {x} by TARSKI:def 1;
   hence x in S by A7,XBOOLE_0:def 5;
   thus S\{x}=f/\S by A7,XBOOLE_1:48
    .=X by A6,XBOOLE_1:28;
  end;
  given x such that
   A8: x in S and
   A9: S\{x}=X;
  reconsider f=X as finite Simplex of K by A9;
  card f=n-1+1 by A5,A8,A9,STIRL2_1:55;
  then A10: f is Simplex of n-1,K by A2,A3,Def18;
  X c=S by A9,XBOOLE_1:36;
  hence thesis by A1,A4,A10,Def19;
 end;
