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 Th55:
  r <> 0 implies Affin (r*AR) = r * Affin AR
 proof
  assume A1: r<>0;
  r"*(r*AR)=(r"*r)*AR by Th10
   .=1*AR by A1,XCMPLX_0:def 7
   .=AR by Th11;
  then AR c=r"*Affin(r*AR) by Lm7,Th9;
  then A2: Affin AR c=r"*Affin(r*AR) by Th51,Th54;
  r*AR c=r*Affin AR by Lm7,Th9;
  then A3: Affin(r*AR)c=r*Affin AR by Th51,Th54;
  r*(r"*Affin(r*AR))=(r*r")*Affin(r*AR) by Th10
   .=1*Affin(r*AR) by A1,XCMPLX_0:def 7
   .=Affin(r*AR) by Th11;
  then r*Affin AR c=Affin(r*AR) by A2,Th9;
  hence thesis by A3;
 end;
