reserve X for AffinPlane;
reserve o,a,a1,a2,a3,a4,b,b1,b2,b3,b4,c,c1,c2,d,d1,d2, d3,d4,d5,e1,e2,x,y,z
  for Element of X;
reserve Y,Z,M,N,A,K,C for Subset of X;
reserve X for OrtAfPl;
reserve o9,a9,a19,a29,a39,a49,b9,b19,b29,b39,b49,c9,c19 for Element of X;
reserve o,a,a1,a2,a3,a4,b,b1,b2,b3,b4,c,c1 for Element of the AffinStruct of X;
reserve M9,N9 for Subset of X;
reserve A,M,N for Subset of the AffinStruct of X;

theorem
  the AffinStruct of X is translational iff X is satisfying_des
proof
  X is satisfying_des implies the AffinStruct of X is translational
  proof
    assume
A1: X is satisfying_des;
    now
      let A,M,N,a,b,c,a1,b1,c1;
      assume that
A2:   A // M and
A3:   A // N and
A4:   a in A and
A5:   a1 in A and
A6:   b in M and
A7:   b1 in M and
A8:   c in N and
A9:   c1 in N and
A10:  A is being_line and
A11:  M is being_line and
A12:  N is being_line and
A13:  A<>M and
A14:  A<>N and
A15:  a,b // a1,b1 and
A16:  a,c // a1,c1;
      reconsider a9=a,a19=a1,b9=b,b19=b1,c9=c,c19=c1 as Element of X;
      b,c // b1,c1
      proof
        assume
A17:    not b,c // b1,c1;
A18:    a<>a1
        proof
          assume
A19:      a=a1;
A20:      c =c1
          proof
            LIN a,c,c1 by A16,A19,AFF_1:def 1;
            then
A21:        LIN c,c1,a by AFF_1:6;
            assume c <>c1;
            then a in N by A8,A9,A12,A21,AFF_1:25;
            hence contradiction by A3,A4,A14,AFF_1:45;
          end;
          b=b1
          proof
            LIN a,b,b1 by A15,A19,AFF_1:def 1;
            then
A22:        LIN b,b1,a by AFF_1:6;
            assume b<>b1;
            then a in M by A6,A7,A11,A22,AFF_1:25;
            hence contradiction by A2,A4,A13,AFF_1:45;
          end;
          hence contradiction by A17,A20,AFF_1:2;
        end;
A23:    not LIN a9,a19,b9 & not LIN a9,a19,c9
        proof
          assume LIN a9,a19,b9 or LIN a9,a19,c9;
          then LIN a,a1,b or LIN a,a1,c by ANALMETR:40;
          then b in A or c in A by A4,A5,A10,A18,AFF_1:25;
          hence contradiction by A2,A3,A6,A8,A13,A14,AFF_1:45;
        end;
        a,a1 // c,c1 by A3,A4,A5,A8,A9,AFF_1:39;
        then
A24:    a9,a19 // c9,c19 by ANALMETR:36;
        a,a1 // b,b1 by A2,A4,A5,A6,A7,AFF_1:39;
        then
A25:    a9,a19 // b9,b19 by ANALMETR:36;
A26:    a9,c9 // a19,c19 by A16,ANALMETR:36;
        a9,b9 // a19,b19 by A15,ANALMETR:36;
        then b9,c9 // b19,c19 by A1,A23,A25,A24,A26,CONMETR:def 8;
        hence contradiction by A17,ANALMETR:36;
      end;
      hence b,c // b1,c1;
    end;
    hence thesis by AFF_2:def 11;
  end;
  hence thesis by CONMETR:8;
end;
