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
  v in I & w in Affin I & p in Affin(I\{v}) & w = r*v + (1-r)*p
    implies r = (w|--I).v
 proof
  assume that
   A1: v in I and
   w in Affin I and
   A2: p in Affin(I\{v}) and
   A3: w=r*v+(1-r)*p;
  A4: I c=conv I by CONVEX1:41;
  Carrier(p|--(I\{v}))c=I\{v} by RLVECT_2:def 6;
  then not v in Carrier(p|--(I\{v})) by ZFMISC_1:56;
  then A5: (p|--(I\{v})).v=0;
  I\{v}c=I by XBOOLE_1:36;
  then Affin(I\{v})c=Affin I & I c=Affin I by Lm7,Th52;
  hence (w|--I).v=((1-r)*(p|--I)+r*(v|--I)).v by A1,A2,A3,Th70
   .=((1-r)*(p|--I)).v+(r*(v|--I)).v by RLVECT_2:def 10
   .=((1-r)*(p|--I)).v+r*((v|--I).v) by RLVECT_2:def 11
   .=(1-r)*((p|--I).v)+r*((v|--I).v) by RLVECT_2:def 11
   .=(1-r)*((p|--I).v)+r*1 by A1,A4,Th72
   .=(1-r)*((p|--(I\{v})).v)+r*1 by A2,Th77,XBOOLE_1:36
   .=r by A5;
 end;
