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 Th69:
  for A,B be Subset of RLS st A c= Affin B holds Affin (A\/B) = Affin B
 proof
  let A,B be Subset of RLS such that
   A1: A c=Affin B;
  set AB={C where C is Affine Subset of RLS:A\/B c=C};
  B c=Affin B by Lm7;
  then A\/B c=Affin B by A1,XBOOLE_1:8;
  then Affin B in AB;
  then A2: Affin(A\/B)c=Affin B by SETFAM_1:3;
  Affin B c=Affin(A\/B) by Th52,XBOOLE_1:7;
  hence thesis by A2;
 end;
