theorem Th15: :: TDGROUP:15

for ADG being Uniquely_Two_Divisible_Group st ex a, b being Element of ADG st a <> b holds
( ex a, b being Element of (AV ADG) st a <> b & ( for a, b, c being Element of (AV ADG) st a,b // c,c holds
a = b ) & ( for a, b, c, d, p, q being Element of (AV ADG) st a,b // p,q & c,d // p,q holds
a,b // c,d ) & ( for a, b, c being Element of (AV ADG) ex d being Element of (AV ADG) st a,b // c,d ) & ( for a, b, c, a9, b9, c9 being Element of (AV ADG) st a,b // a9,b9 & a,c // a9,c9 holds
b,c // b9,c9 ) & ( for a, c being Element of (AV ADG) ex b being Element of (AV ADG) st a,b // b,c ) & ( for a, b, c, b9 being Element of (AV ADG) st a,b // b,c & a,b9 // b9,c holds
b = b9 ) & ( for a, b, c, d being Element of (AV ADG) st a,b // c,d holds
a,c // b,d ) )