reserve ADG for Uniquely_Two_Divisible_Group;
reserve a,b,c,d,a9,b9,c9,p,q for Element of ADG;
reserve x,y for set;

theorem Th9:
  a,b ==> p,q & c,d ==> p,q implies a,b ==> c,d
proof
  assume that
A1: a,b ==> p,q and
A2: c,d ==> p,q;
  a # q = b # p by A1,Th5;
  then a + (q + d) = (b + p) + d by RLVECT_1:def 3
    .= b + (p + d) by RLVECT_1:def 3
    .= b + (c + q) by A2,Th5;
  then (a + d) + q = b + (c + q) by RLVECT_1:def 3
    .= (b + c) + q by RLVECT_1:def 3;
  then a + d = b + c by RLVECT_1:8;
  hence thesis by Th5;
end;
