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;
