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 Th54:
  AR is Affine implies r * AR is Affine
 proof
  assume A1: AR is Affine;
  let v1,v2 be VECTOR of R,s;
  assume v1 in r*AR;
  then consider w1 be Element of R such that
   A2: v1=r*w1 and
   A3: w1 in AR;
  assume v2 in r*AR;
  then consider w2 be Element of R such that
   A4: v2=r*w2 and
   A5: w2 in AR;
  A6: (1-s)*w1+s*w2 in AR by A1,A3,A5;
  A7: (1-s)*(r*w1)=((1-s)*r)*w1 by RLVECT_1:def 7
   .=r*((1-s)*w1) by RLVECT_1:def 7;
  s*(r*w2)=(s*r)*w2 by RLVECT_1:def 7
   .=r*(s*w2) by RLVECT_1:def 7;
  then (1-s)*v1+s*v2=r*((1-s)*w1+s*w2) by A2,A4,A7,RLVECT_1:def 5;
  hence thesis by A6;
 end;
