reserve T for non empty Abelian
  add-associative right_zeroed right_complementable RLSStruct,
  X,Y,Z,B,C,B1,B2 for Subset of T,
  x,y,p for Point of T;
reserve t,s,r1 for Real;
reserve n for Element of NAT;
reserve X,Y,B1,B2 for Subset of TOP-REAL n;
reserve x,y for Point of TOP-REAL n;

theorem
  t<>0 implies t(.)(X (O) Y) = t(.)X (O) t(.)Y
proof
  assume
A1: t<>0;
  t(.)(X (O) Y) =t(.)(X (-) Y) (+) t(.)Y by Th63
    .=t(.)X (-) t(.)Y (+) t(.)Y by A1,Th64;
  hence thesis;
end;
