reserve x,y for set,
        r,s for Real,
        S for non empty addLoopStr,
        LS,LS1,LS2 for Linear_Combination of S,
        G for Abelian add-associative right_zeroed right_complementable
          non empty addLoopStr,
        LG,LG1,LG2 for Linear_Combination of G,
        g,h for Element of G,
        RLS for non empty RLSStruct,
        R for vector-distributive scalar-distributive scalar-associative
        scalar-unitalnon empty RLSStruct,
        AR for Subset of R,
        LR,LR1,LR2 for Linear_Combination of R,
        V for RealLinearSpace,
        v,v1,v2,w,p for VECTOR of V,
        A,B for Subset of V,
        F1,F2 for Subset-Family of V,
        L,L1,L2 for Linear_Combination of V;
reserve I for affinely-independent Subset of V;

theorem
  for B st x in conv I & for y st y in B holds (x|--I).y = 0
    holds x in conv (I\B)
 proof
  let B such that
   A1: x in conv I and
   A2: for y st y in B holds(x|--I).y=0;
  set IB=I\B;
  A3: conv I c=Affin I by Th65;
  then x|--I=x|--IB by A1,A2,Th75;
  then A4: for v st v in IB holds 0<=(x|--IB).v by A1,Th71;
  A5: IB is affinely-independent by Th43,XBOOLE_1:36;
  x in Affin IB by A1,A2,A3,Th75;
  hence thesis by A4,A5,Th73;
 end;
