let AS be AffinSpace; :: thesis: ( AS is not AffinPlane implies AS is translational )

assume AS is not AffinPlane ; :: thesis: AS is translational

then for A, P, C being Subset of AS

for a, b, c, a9, b9, c9 being Element of AS st A // P & A // C & a in A & a9 in A & b in P & b9 in P & c in C & c9 in C & A is being_line & P is being_line & C is being_line & A <> P & A <> C & a,b // a9,b9 & a,c // a9,c9 holds

b,c // b9,c9 by Th51;

hence AS is translational by AFF_2:def 11; :: thesis: verum

assume AS is not AffinPlane ; :: thesis: AS is translational

then for A, P, C being Subset of AS

for a, b, c, a9, b9, c9 being Element of AS st A // P & A // C & a in A & a9 in A & b in P & b9 in P & c in C & c9 in C & A is being_line & P is being_line & C is being_line & A <> P & A <> C & a,b // a9,b9 & a,c // a9,c9 holds

b,c // b9,c9 by Th51;

hence AS is translational by AFF_2:def 11; :: thesis: verum