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
  A\/{v} is affinely-independent iff
    A is affinely-independent & (v in A or not v in Affin A)
 proof
  set Av=A\/{v};
  v in {v} by TARSKI:def 1;
  then A1: v in Av by XBOOLE_0:def 3;
  A2: A c=Av by XBOOLE_1:7;
  hereby assume A3: Av is affinely-independent;
   hence A is affinely-independent by Th43,XBOOLE_1:7;
   v in Affin A implies v in A
   proof
    assume v in Affin A;
    then {v}c=Affin A by ZFMISC_1:31;
    then Affin Av=Affin A by Th69;
    hence thesis by A2,A1,A3,Th58;
   end;
   hence v in A or not v in Affin A;
  end;
  assume that
   A4: A is affinely-independent and
   A5: v in A or not v in Affin A;
  per cases by A5;
  suppose v in A;
   hence thesis by A4,ZFMISC_1:40;
  end;
  suppose A6: not v in Affin A & not v in A;
   consider I be affinely-independent Subset of V such that
    A7: A c=I and
    A8: I c=Av and
    A9: Affin I=Affin Av by A2,A4,Th60;
   assume A10: not Av is affinely-independent;
   not v in I
   proof
    assume v in I;
    then {v}c=I by ZFMISC_1:31;
    hence contradiction by A7,A10,Th43,XBOOLE_1:8;
   end;
   then A11: I c=Av\{v} by A8,ZFMISC_1:34;
   A12: Av c=Affin Av by Lm7;
   Av\{v}=A by A6,ZFMISC_1:117;
   then I=A by A7,A11;
   hence contradiction by A1,A6,A9,A12;
  end;
 end;
