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 Th73:
  for I st x in Affin I & for v st v in I holds 0 <= (x|--I).v
    holds x in conv I
 proof
  let I such that
   A1: x in Affin I and
   A2: for v st v in I holds 0<=(x|--I).v;
  set xI=x|--I;
  A3: Sum xI=x by A1,Def7;
  reconsider I1=I as non empty Subset of V by A1;
  A4: for v holds 0<=xI.v
  proof
   let v;
   A5: v in Carrier xI or not v in Carrier xI;
   Carrier xI c=I by RLVECT_2:def 6;
   hence thesis by A2,A5;
  end;
  sum xI=1 by A1,Def7;
  then A6: xI is convex by A4,Th62;
  then conv(I1)={Sum(L) where L is Convex_Combination of I1:L in ConvexComb(V)}
& xI in ConvexComb(V) by CONVEX3:5,def 1;
  hence thesis by A3,A6;
 end;
