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 Th77:
  B c= I & x in Affin B implies x |-- B = x |-- I
 proof
  assume that
   A1: B c=I and
   A2: x in Affin B;
  B is affinely-independent by A1,Th43;
  then A3: Sum(x|--B)=x & sum(x|--B)=1 by A2,Def7;
  Affin B c=Affin I & x|--B is Linear_Combination of I by A1,Th52,RLVECT_2:21;
  hence thesis by A2,A3,Def7;
 end;
