 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;
reserve EV for Enumeration of Affv,
        EN for Enumeration of Affn;
reserve pnA for Element of(TOP-REAL n)|Affin Affn;

theorem Th29:
  for A be Subset of V st not x in A holds |--(A,x) = [#]V-->0
proof
  let A be Subset of V;
  set Ax=|--(A,x);
  assume A1: not x in A;
  A2: now let y be object;
   assume y in dom Ax;
   then A3: Ax.y=(y|--A).x by Def3;
   Carrier(y|--A)c=A by RLVECT_2:def 6;
   then A4: not x in Carrier(y|--A) by A1;
   per cases;
   suppose x in [#]V;
    hence Ax.y=0 by A3,A4,RLVECT_2:19;
   end;
   suppose not x in [#]V;
    then not x in dom(y|--A);
    hence Ax.y=0 by A3,FUNCT_1:def 2;
   end;
  end;
  dom Ax=[#]V by FUNCT_2:def 1;
  hence thesis by A2,FUNCOP_1:11;
 end;
