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 Th53:
  Affin (v+A) = v + Affin A
 proof
  v+A c=v+Affin A by Lm7,RLTOPSP1:8;
  then A1: Affin(v+A)c=v+Affin A by Th51,RUSUB_4:31;
  -v+(v+A)=(-v+v)+A by Th5
   .=0.V+A by RLVECT_1:5
   .=A by Th6;
  then A c=-v+Affin(v+A) by Lm7,RLTOPSP1:8;
  then A2: Affin A c=-v+Affin(v+A) by Th51,RUSUB_4:31;
  v+(-v+Affin(v+A))=(v+-v)+Affin(v+A) by Th5
   .=0.V+Affin(v+A) by RLVECT_1:5
   .=Affin(v+A) by Th6;
  then v+Affin A c=Affin(v+A) by A2,RLTOPSP1:8;
  hence thesis by A1;
 end;
