reserve M for non empty MetrSpace,
        F,G for open Subset-Family of TopSpaceMetr M;
reserve L for Lebesgue_number of F;
reserve n,k for Nat,
        r for Real,
        X for set,
        M for Reflexive non empty MetrStruct,
        A for Subset of M,
        K for SimplicialComplexStr;
reserve V for RealLinearSpace,
        Kv for non void SimplicialComplex of V;
reserve A for Subset of TOP-REAL n;
reserve A for affinely-independent Subset of TOP-REAL n;

theorem
  for A st card A = n+1
  for f be continuous Function of (TOP-REAL n)|conv A,(TOP-REAL n)|conv A
    holds f is with_fixpoint
 proof
  let A be affinely-independent Subset of TOP-REAL n such that
   A1: card A=n+1;
  let f be continuous Function of(TOP-REAL n)|conv A,(TOP-REAL n)|conv A;
  consider x be Point of TOP-REAL n such that
   A2: x in dom f & f.x=x by A1,Th23;
  x is_a_fixpoint_of f by A2,ABIAN:def 3;
  hence thesis by ABIAN:def 5;
 end;
