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;
