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 Th11:
  a,b ==> a9,b9 & a,c ==> a9,c9 implies b,c ==> b9,c9
proof
  assume a,b ==> a9,b9 & a,c ==> a9,c9;
  then a + b9 = b + a9 & a + c9= c + a9 by Th5;
  then b + (a9 + (a + c9)) = (c + a9) + (a + b9) by RLVECT_1:def 3
    .= c + (a9 + (a + b9)) by RLVECT_1:def 3;
  then b + ((a9 + a) + c9) = c + (a9 + (a + b9)) by RLVECT_1:def 3
    .= c + ((a9 + a) + b9) by RLVECT_1:def 3;
  then (b + c9) + (a9 + a) = c + (b9 + (a9 + a)) by RLVECT_1:def 3
    .= (c + b9) + (a9 + a) by RLVECT_1:def 3;
  then b + c9 = c + b9 by RLVECT_1:8;
  hence thesis by Th5;
end;
