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
  x in I implies (conv I)\{x} is convex
 proof
  assume A1: x in I;
  then reconsider X=x as Element of V;
  A2: conv I c=Affin I by Th65;
  now let v1,v2,r;
   set rv12=r*v1+(1-r)*v2;
   assume that
    A3: 0<r and
    A4: r<1;
   assume that
    A5: v1 in (conv I)\{x} and
    A6: v2 in (conv I)\{x};
   A7: 1-r>1-1 by A4,XREAL_1:10;
   A8: v2 in conv I by A6,ZFMISC_1:56;
   then A9: (v2|--I).X<=1 by Th71;
   A10: v1 in conv I by A5,ZFMISC_1:56;
   then A11: (v1|--I).X<=1 by Th71;
   A12: rv12|--I=(1-r)*(v2|--I)+r*(v1|--I) by A2,A10,A8,Th70;
   A13: now let w;
    assume w in I;
    A14: (rv12|--I).w=((1-r)*(v2|--I)).w+(r*(v1|--I)).w by A12,RLVECT_2:def 10
     .=(1-r)*((v2|--I).w)+(r*(v1|--I)).w by RLVECT_2:def 11
     .=(1-r)*((v2|--I).w)+r*((v1|--I).w) by RLVECT_2:def 11;
    (v2|--I).w>=0 & (v1|--I).w>=0 by A10,A8,Th71;
    hence 0<=(rv12|--I).w by A3,A7,A14;
   end;
   rv12 in Affin I by A2,A10,A8,RUSUB_4:def 4;
   then A15: rv12 in conv I by A13,Th73;
   v1<>x by A5,ZFMISC_1:56;
   then (v1|--I).X<>1 by A10,Th72;
   then (v1|--I).X<1 by A11,XXREAL_0:1;
   then A16: r*((v1|--I).X)<r*1 by A3,XREAL_1:68;
   v2<>x by A6,ZFMISC_1:56;
   then (v2|--I).X<>1 by A8,Th72;
   then (v2|--I).X<1 by A9,XXREAL_0:1;
   then (1-r)*((v2|--I).X)<(1-r)*1 by A7,XREAL_1:68;
   then A17: (1-r)*((v2|--I).X)+r*((v1|--I).X)<(1-r)*1+r*1 by A16,XREAL_1:8;
   (rv12|--I).X=((1-r)*(v2|--I)).X+(r*(v1|--I)).X by A12,RLVECT_2:def 10
    .=(1-r)*((v2|--I).X)+(r*(v1|--I)).X by RLVECT_2:def 11
    .=(1-r)*((v2|--I).X)+r*((v1|--I).X) by RLVECT_2:def 11;
   then rv12<>X by A1,A15,A17,Th72;
   hence rv12 in (conv I)\{x} by A15,ZFMISC_1:56;
  end;
  hence thesis;
 end;
