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 Th71:
  x in conv I implies x|--I is convex & 0 <= (x|--I).v & (x|--I).v <= 1
 proof
  assume A1: x in conv I;
  then reconsider I1=I as non empty Subset of V;
  conv(I1)={Sum(L) where L is Convex_Combination of I1:L in ConvexComb(V)}
by CONVEX3:5;
  then consider L be Convex_Combination of I1 such that
   A2: Sum L=x and
   L in ConvexComb(V) by A1;
  conv I c=Affin I & sum L=1 by Th62,Th65;
  then L=x|--I by A1,A2,Def7;
  hence thesis by Th62,Th63;
 end;
