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
  X is satisfying_DES iff the AffinStruct of X is Desarguesian
proof
A1: X is satisfying_DES implies the AffinStruct of X is Desarguesian
  proof
    assume
A2: X is satisfying_DES;
    now
      let A,M,N,o,a,b,c,a1,b1,c1;
      assume that
A3:   o in A and
A4:   o in M and
A5:   o in N and
A6:   o<>a and
A7:   o<>b and
A8:   o<>c and
A9:   a in A and
A10:  a1 in A and
A11:  b in M and
A12:  b1 in M and
A13:  c in N and
A14:  c1 in N and
A15:  A is being_line and
A16:  M is being_line and
A17:  N is being_line and
A18:  A<>M and
A19:  A<>N and
A20:  a,b // a1,b1 and
A21:  a,c // a1,c1;
      reconsider o9=o,a9=a,a19=a1,b9=b,b19=b1,c9=c,c19=c1 as Element of X;
      o,b // o,b1 by A4,A11,A12,A16,AFF_1:39,41;
      then
A22:  LIN o,b,b1 by AFF_1:def 1;
      then
A23:  LIN o9,b9,b19 by ANALMETR:40;
      o,a // o,a1 by A3,A9,A10,A15,AFF_1:39,41;
      then
A24:  LIN o,a, a1 by AFF_1:def 1;
      o,c // o,c1 by A5,A13,A14,A17,AFF_1:39,41;
      then
A25:  LIN o,c,c1 by AFF_1:def 1;
      then
A26:  LIN o9,c9,c19 by ANALMETR:40;
      assume
A27:  not b,c // b1,c1;
A28:  not LIN b9,b19,a9 & not LIN a9,a19,c9
      proof
A29:    a<>a1
        proof
          assume
A30:      a=a1;
A31:      c =c1
          proof
A32:        not LIN o,a,c
            proof
              assume LIN o,a,c;
              then c in A by A3,A6,A9,A15,AFF_1:25;
              then
A33:          o,c // A by A3,A15,AFF_1:52;
              o,c // N by A5,A13,A17,AFF_1:52;
              hence contradiction by A3,A5,A8,A19,A33,AFF_1:45,53;
            end;
            assume c <>c1;
            hence contradiction by A21,A25,A30,A32,AFF_1:14;
          end;
          b=b1
          proof
A34:        not LIN o,a,b
            proof
              assume LIN o,a,b;
              then b in A by A3,A6,A9,A15,AFF_1:25;
              then
A35:          o,b // A by A3,A15,AFF_1:52;
              o,b // M by A4,A11,A16,AFF_1:52;
              hence contradiction by A3,A4,A7,A18,A35,AFF_1:45,53;
            end;
            assume b<>b1;
            hence contradiction by A20,A22,A30,A34,AFF_1:14;
          end;
          hence contradiction by A27,A31,AFF_1:2;
        end;
A36:    b<>b1
        proof
A37:      not LIN o,b,a
          proof
            assume LIN o,b,a;
            then a in M by A4,A7,A11,A16,AFF_1:25;
            then
A38:        o,a // M by A4,A16,AFF_1:52;
            o,a // A by A3,A9,A15,AFF_1:52;
            hence contradiction by A3,A4,A6,A18,A38,AFF_1:45,53;
          end;
          assume b=b1;
          then b,a // b,a1 by A20,AFF_1:4;
          hence contradiction by A24,A29,A37,AFF_1:14;
        end;
        assume LIN b9,b19,a9 or LIN a9,a19,c9;
        then LIN b,b1,a or LIN a,a1,c by ANALMETR:40;
        then a in M or c in A by A9,A10,A11,A12,A15,A16,A29,A36,AFF_1:25;
        then
A39:    o,a // M or o,c // A by A3,A4,A15,A16,AFF_1:52;
A40:    o,c // N by A5,A13,A17,AFF_1:52;
        o,a // A by A3,A9,A15,AFF_1:52;
        hence contradiction by A3,A4,A5,A6,A8,A18,A19,A39,A40,AFF_1:45,53;
      end;
A41:  a9,c9 // a19,c19 by A21,ANALMETR:36;
A42:  a9,b9 // a19,b19 by A20,ANALMETR:36;
A43:  o9<>a19 & o9<>b19 & o9<>c19
      proof
A44:    now
          assume
A45:      o9=a19;
A46:      o=c1
          proof
A47:        not LIN c,a,o
            proof
              assume LIN c,a,o;
              then LIN o,a,c by AFF_1:6;
              then c in A by A3,A6,A9,A15,AFF_1:25;
              then
A48:          o,c // A by A3,A15,AFF_1:52;
              o,c // N by A5,A13,A17,AFF_1:52;
              hence contradiction by A3,A5,A8,A19,A48,AFF_1:45,53;
            end;
            assume
A49:        o<>c1;
A50:        o,c // o,c1 by A5,A13,A14,A17,AFF_1:39,41;
            then a,c // o,c by A21,A45,A49,AFF_1:5;
            then c,a // c,o by AFF_1:4;
            then LIN c,a,o by AFF_1:def 1;
            then LIN o,a,c by AFF_1:6;
            then o,a // o,c by AFF_1:def 1;
            then o,a // o,c1 by A8,A50,AFF_1:5;
            then LIN o,a,c1 by AFF_1:def 1;
            then LIN a,o,c1 by AFF_1:6;
            then
A51:        a,o // a,c1 by AFF_1:def 1;
            LIN c,o,c1 by A25,AFF_1:6;
            hence contradiction by A49,A47,A51,AFF_1:14;
          end;
          o=b1
          proof
A52:        not LIN b,a,o
            proof
              assume LIN b,a,o;
              then LIN o,a,b by AFF_1:6;
              then b in A by A3,A6,A9,A15,AFF_1:25;
              then
A53:          o,b // A by A3,A15,AFF_1:52;
              o,b // M by A4,A11,A16,AFF_1:52;
              hence contradiction by A3,A4,A7,A18,A53,AFF_1:45,53;
            end;
            assume
A54:        o<>b1;
A55:        o,b1 // o,b by A4,A11,A12,A16,AFF_1:39,41;
            then a,b // o,b by A20,A45,A54,AFF_1:5;
            then b,a // b,o by AFF_1:4;
            then LIN b,a,o by AFF_1:def 1;
            then LIN o,a,b by AFF_1:6;
            then o,a // o,b by AFF_1:def 1;
            then o,a // o,b1 by A7,A55,AFF_1:5;
            then LIN o,a,b1 by AFF_1:def 1;
            then LIN a,o,b1 by AFF_1:6;
            then
A56:        a,o // a,b1 by AFF_1:def 1;
            LIN b,o,b1 by A22,AFF_1:6;
            hence contradiction by A54,A52,A56,AFF_1:14;
          end;
          hence contradiction by A27,A46,AFF_1:3;
        end;
A57:    now
A58:      not LIN a,b,o
          proof
            assume LIN a,b,o;
            then LIN o,a,b by AFF_1:6;
            then b in A by A3,A6,A9,A15,AFF_1:25;
            then
A59:        o,b // A by A3,A15,AFF_1:52;
            o,b // M by A4,A11,A16,AFF_1:52;
            hence contradiction by A3,A4,A7,A18,A59,AFF_1:45,53;
          end;
A60:      a1,o // a,o by A3,A9,A10,A15,AFF_1:39,41;
          assume o9=b19;
          then a,b // a,o by A20,A44,A60,AFF_1:5;
          then LIN a,b,o by AFF_1:def 1;
          then LIN o,b,a by AFF_1:6;
          then o,b // o,a by AFF_1:def 1;
          then o,b // a,o by AFF_1:4;
          then o,b // a1,o by A6,A60,AFF_1:5;
          then o,b // o,a1 by AFF_1:4;
          then LIN o,b,a1 by AFF_1:def 1;
          then LIN b,o,a1 by AFF_1:6;
          then
A61:      b,o // b,a1 by AFF_1:def 1;
          LIN a,o,a1 by A24,AFF_1:6;
          hence contradiction by A44,A58,A61,AFF_1:14;
        end;
A62:    now
A63:      not LIN a,c,o
          proof
            assume LIN a,c,o;
            then LIN o,a,c by AFF_1:6;
            then c in A by A3,A6,A9,A15,AFF_1:25;
            then
A64:        o,c // A by A3,A15,AFF_1:52;
            o,c // N by A5,A13,A17,AFF_1:52;
            hence contradiction by A3,A5,A8,A19,A64,AFF_1:45,53;
          end;
A65:      a1,o // a,o by A3,A9,A10,A15,AFF_1:39,41;
          assume o9=c19;
          then a,c // a,o by A21,A44,A65,AFF_1:5;
          then LIN a,c,o by AFF_1:def 1;
          then LIN o,c,a by AFF_1:6;
          then o,c // o,a by AFF_1:def 1;
          then o,c // a,o by AFF_1:4;
          then o,c // a1,o by A6,A65,AFF_1:5;
          then o,c // o,a1 by AFF_1:4;
          then LIN o,c,a1 by AFF_1:def 1;
          then LIN c,o,a1 by AFF_1:6;
          then
A66:      c,o // c,a1 by AFF_1:def 1;
          LIN a,o,a1 by A24,AFF_1:6;
          hence contradiction by A44,A63,A66,AFF_1:14;
        end;
        assume not thesis;
        hence contradiction by A44,A57,A62;
      end;
      LIN o9,a9,a19 by A24,ANALMETR:40;
      then b9,c9 // b19,c19 by A2,A6,A7,A8,A23,A26,A43,A28,A42,A41,
CONAFFM:def 1;
      hence contradiction by A27,ANALMETR:36;
    end;
    hence thesis by AFF_2:def 4;
  end;
  the AffinStruct of X is Desarguesian implies X is satisfying_DES
  proof
    assume
A67: the AffinStruct of X is Desarguesian;
    now
      let o9,a9,a19,b9,b19,c9,c19;
      assume that
A68:  o9<>a9 and
      o9<>a19 and
A69:  o9<>b9 and
      o9<>b19 and
A70:  o9<>c9 and
      o9<>c19 and
A71:  not LIN b9,b19,a9 and
A72:  not LIN a9,a19,c9 and
A73:  LIN o9,a9,a19 and
A74:  LIN o9,b9,b19 and
A75:  LIN o9,c9,c19 and
A76:  a9,b9 // a19,b19 and
A77:  a9,c9 // a19,c19;
      reconsider o=o9,a=a9,a1=a19,b=b9,b1=b19,c =c9,c1=c19
        as Element of the AffinStruct of X;
A78:  not LIN a,a1,c by A72,ANALMETR:40;
A79:  a,c // a1,c1 by A77,ANALMETR:36;
A80:  a,b // a1,b1 by A76,ANALMETR:36;
      LIN o,b,b1 by A74,ANALMETR:40;
      then consider M such that
A81:  M is being_line and
A82:  o in M and
A83:  b in M and
A84:  b1 in M by AFF_1:21;
      LIN o,c,c1 by A75,ANALMETR:40;
      then consider N such that
A85:  N is being_line and
A86:  o in N and
A87:  c in N and
A88:  c1 in N by AFF_1:21;
      LIN o,a,a1 by A73,ANALMETR:40;
      then consider A such that
A89:  A is being_line and
A90:  o in A and
A91:  a in A and
A92:  a1 in A by AFF_1:21;
A93:  not LIN b,b1,a by A71,ANALMETR:40;
      A<>M & A<>N
      proof
        assume not thesis;
        then
        b,b1 // b,a or a,a1 // a,c by A89,A91,A92,A83,A84,A87,AFF_1:39,41;
        hence contradiction by A93,A78,AFF_1:def 1;
      end;
      then b,c // b1,c1 by A67,A68,A69,A70,A89,A90,A91,A92,A81,A82,A83,A84,A85
,A86,A87,A88,A80,A79,AFF_2:def 4;
      hence b9,c9 // b19,c19 by ANALMETR:36;
    end;
    hence thesis by CONAFFM:def 1;
  end;
  hence thesis by A1;
end;
