theorem Th77:
  B c= I & x in Affin B implies x |-- B = x |-- I
 proof
  assume that
   A1: B c=I and
   A2: x in Affin B;
  B is affinely-independent by A1,Th43;
  then A3: Sum(x|--B)=x & sum(x|--B)=1 by A2,Def7;
  Affin B c=Affin I & x|--B is Linear_Combination of I by A1,Th52,RLVECT_2:21;
  hence thesis by A2,A3,Def7;
 end;
