theorem Th18:
  v in If implies ((center_of_mass V).If |-- If).v = 1/card If
  proof
    consider L be Linear_Combination of If such that
    A1: Sum L=1/card If*Sum If & sum L=1/card If*card If and
    A2: L=(ZeroLC V)+*(If-->1/card If) by Th15;
    assume A3: v in If;
    then A4: conv If c=Affin If & (center_of_mass V).If in conv If
      by Th16,RLAFFIN1:65;
    (center_of_mass V).If=Sum L & sum L=1 by A1,A3,Def2,XCMPLX_1:87;
    then dom(If-->1/card If)=If & L=(center_of_mass V).If|--If
      by A4,RLAFFIN1:def 7;
    hence ((center_of_mass V).If|--If).v=(If-->1/card If).v by A2,A3,FUNCT_4:13
    .=1/card If by A3,FUNCOP_1:7;
  end;
