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 Th70:
  v1 in Affin I & v2 in Affin I implies
  ((1-r)*v1+r*v2) |-- I = (1-r) * (v1|--I) + r * (v2|--I)
 proof
  assume that
   A1: v1 in Affin I and
   A2: v2 in Affin I;
  set rv12=(1-r)*v1+r*v2;
  A3: rv12 in Affin I by A1,A2,RUSUB_4:def 4;
  (1-r)*(v1|--I) is Linear_Combination of I & r*(v2|--I) is Linear_Combination
of I by RLVECT_2:44;
  then A4: (1-r)*(v1|--I)+r*(v2|--I) is Linear_Combination of I by RLVECT_2:38;
  A5: Sum((1-r)*(v1|--I)+r*(v2|--I))=Sum((1-r)*(v1|--I))+Sum(r*(v2|--I)) by
RLVECT_3:1
   .=(1-r)*Sum(v1|--I)+Sum(r*(v2|--I)) by RLVECT_3:2
   .=(1-r)*Sum(v1|--I)+r*Sum(v2|--I) by RLVECT_3:2
   .=(1-r)*v1+r*Sum(v2|--I) by A1,Def7
   .=rv12 by A2,Def7;
  sum((1-r)*(v1|--I)+r*(v2|--I))=sum((1-r)*(v1|--I))+sum(r*(v2|--I)) by Th34
   .=(1-r)*sum(v1|--I)+sum(r*(v2|--I)) by Th35
   .=(1-r)*sum(v1|--I)+r*sum(v2|--I) by Th35
   .=(1-r)*1+r*sum(v2|--I) by A1,Def7
   .=(1-r)*1+r*1 by A2,Def7
   .=1;
  hence thesis by A3,A4,A5,Def7;
 end;
