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
  Affin (r*A) = r * Affin A
 proof
  per cases;
  suppose A1: r=0;
   then A2: r*Affin A c={0.V} by Th12;
   A3: r*Affin A c=r*A
   proof
    let x be object;
    assume A4: x in r*Affin A;
    then ex v be Element of V st x=r*v & v in Affin A;
    then A is non empty;
    then consider v be object such that
     A5: v in A;
    reconsider v as Element of V by A5;
    A6: r*v in r*A by A5;
    x=0.V by A2,A4,TARSKI:def 1;
    hence thesis by A1,A6,RLVECT_1:10;
   end;
   r*A c={0.V} by A1,Th12;
   then A7: r*A is empty or r*A={0.V} by ZFMISC_1:33;
   {0.V} is Affine by RUSUB_4:23;
   then A8: Affin(r*A)=r*A by A7,Lm8;
   r*A c=r*Affin A by Lm7,Th9;
   hence thesis by A3,A8;
  end;
  suppose r<>0;
   hence thesis by Th55;
  end;
 end;
