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 Th57:
  v in Affin A implies Affin A = v + Up Lin (-v+A)
 proof
  set vA=-v+A;
  set BB={B where B is Affine Subset of V:vA c=B};
  A1: -v+A c=Up(Lin(-v+A))
  proof
   let x be object;
   assume x in -v+A;
   then x in Lin(-v+A) by RLVECT_3:15;
   hence thesis;
  end;
  Up(Lin vA) is Affine by RUSUB_4:24;
  then A2: Up(Lin vA) in BB by A1;
  then A3: Affin vA c=Up(Lin vA) by SETFAM_1:3;
  assume v in Affin A;
  then -v+v in -v+Affin A;
  then A4: 0.V in -v+Affin A by RLVECT_1:5;
  now let Y be set;
   A5: Affin vA=-v+Affin A by Th53;
   assume Y in BB;
   then consider B be Affine Subset of V such that
    A6: Y=B and
    A7: vA c=B;
   A8: Lin vA is Subspace of Lin B by A7,RLVECT_3:20;
   Affin vA c=B by A7,Th51;
   then B=the carrier of Lin B by A4,A5,RUSUB_4:26;
   hence Up(Lin vA)c=Y by A6,A8,RLSUB_1:def 2;
  end;
  then Up(Lin(-v+A))c=Affin vA by A2,SETFAM_1:5;
  then Up(Lin(-v+A))=Affin vA by A3;
  hence v+Up(Lin(-v+A))=v+(-v+Affin A) by Th53
   .=(v+-v)+Affin A by Th5
   .=0.V+Affin A by RLVECT_1:5
   .=Affin A by Th6;
 end;
