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 Th78:
  v1 in Affin A & v2 in Affin A & r+s = 1 implies r*v1 + s*v2 in Affin A
 proof
  A1: Affin A={Sum L where L is Linear_Combination of A:sum L=1} by Th59;
  assume v1 in Affin A;
  then consider L1 be Linear_Combination of A such that
   A2: v1=Sum L1 and
   A3: sum L1=1 by A1;
  assume v2 in Affin A;
  then consider L2 be Linear_Combination of A such that
   A4: v2=Sum L2 and
   A5: sum L2=1 by A1;
  A6: Sum(r*L1+s*L2)=Sum(r*L1)+Sum(s*L2) by RLVECT_3:1
   .=r*Sum L1+Sum(s*L2) by RLVECT_3:2
   .=r*v1+s*v2 by A2,A4,RLVECT_3:2;
  r*L1 is Linear_Combination of A & s*L2 is Linear_Combination of A by
RLVECT_2:44;
  then A7: r*L1+s*L2 is Linear_Combination of A by RLVECT_2:38;
  assume A8: r+s=1;
  sum(r*L1+s*L2)=sum(r*L1)+sum(s*L2) by Th34
   .=r*sum L1+sum(s*L2) by Th35
   .=r*1+s*1 by A3,A5,Th35
   .=1 by A8;
  hence thesis by A1,A7,A6;
 end;
