 reserve x for set,
         n,m,k for Nat,
         r for Real,
         V for RealLinearSpace,
         v,u,w,t for VECTOR of V,
         Av for finite Subset of V,
         Affv for finite affinely-independent Subset of V;
reserve pn for Point of TOP-REAL n,
        An for Subset of TOP-REAL n,
        Affn for affinely-independent Subset of TOP-REAL n,
        Ak for Subset of TOP-REAL k;

theorem Th9:
  for A be affinely-independent Subset of V, B be Subset of V st B c= A
  holds conv A/\Affin B = conv B
proof
  let A be affinely-independent Subset of V;
  let B be Subset of V;
  A1: conv B c=Affin B by RLAFFIN1:65;
  assume A2: B c=A;
  thus conv A/\Affin B c=conv B
  proof
   let x be object;
   assume A3: x in conv A/\Affin B;
   then A4: x in Affin B by XBOOLE_0:def 4;
   A5: x in conv A by A3,XBOOLE_0:def 4;
   A6: now let v be Element of V;
    x|--B=x|--A by A2,A4,RLAFFIN1:77;
    hence v in B implies 0<=(x|--B).v by A5,RLAFFIN1:71;
   end;
   B is affinely-independent by A2,RLAFFIN1:43;
   hence thesis by A4,A6,RLAFFIN1:73;
  end;
  conv B c=conv A by A2,RLAFFIN1:3;
  hence thesis by A1,XBOOLE_1:19;
end;
