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 Th72:
  x in conv I implies ((x|--I).y = 1 iff x = y & x in I)
 proof
  assume A1: x in conv I;
  then reconsider v=x as Element of V;
  hereby assume A2: (x|--I).y=1;
   x|--I is convex by A1,Th71;
   then A3: Carrier(x|--I)={y} by A2,Th64;
   y in {y} by TARSKI:def 1;
   then reconsider v=y as Element of V by A3;
   conv I c=Affin I by Th65;
   hence A4: x=Sum(x|--I) by A1,Def7
    .=(x|--I).v*v by A3,RLVECT_2:35
    .=y by A2,RLVECT_1:def 8;
   Carrier(x|--I)c=I & v in Carrier(x|--I) by A2,RLVECT_2:def 6;
   hence x in I by A4;
  end;
  assume that
   A5: x=y and
   A6: x in I;
  consider L be Linear_Combination of{v} such that
   A7: L.v=jj by RLVECT_4:37;
  Carrier L c={v} by RLVECT_2:def 6;
  then A8: sum L=1 by A7,Th32;
  A9: I c=Affin I by Lm7;
  {v}c=I by A6,ZFMISC_1:31;
  then A10: L is Linear_Combination of I by RLVECT_2:21;
  Sum L=1*v by A7,RLVECT_2:32
   .=v by RLVECT_1:def 8;
  hence thesis by A5,A6,A7,A8,A9,A10,Def7;
 end;
