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;

theorem
  X is Desarguesian iff X is satisfying_Scherungssatz
proof
A1: X is satisfying_Scherungssatz implies X is Desarguesian
  proof
    assume
A2: X is satisfying_Scherungssatz;
    now
      let Y,Z,M,o,a,b,c,a1,b1,c1;
      assume that
A3:   o in Y and
A4:   o in Z and
A5:   o in M and
A6:   o<>a and
A7:   o<>b and
A8:   o<>c and
A9:   a in Y and
A10:  a1 in Y and
A11:  b in Z and
A12:  b1 in Z and
A13:  c in M and
A14:  c1 in M and
A15:  Y is being_line and
A16:  Z is being_line and
A17:  M is being_line and
A18:  Y<>Z and
A19:  Y<>M and
A20:  a,b // a1,b1 and
A21:  a,c // a1,c1;
      assume
A22:  not b,c // b1,c1;
A23:  now
        let Y,Z,M,o,a,b,c,a1,b1,c1,d;
        assume
A24:    X is satisfying_Scherungssatz;
        assume that
A25:    o in Y and
A26:    o in Z and
A27:    o in M and
A28:    o<>a and
A29:    o<>b and
A30:    o<>c and
A31:    a in Y and
A32:    a1 in Y and
A33:    b in Z and
A34:    b1 in Z and
A35:    c in M and
A36:    c1 in M and
A37:    Y is being_line and
A38:    Z is being_line and
A39:    M is being_line and
A40:    Y<>Z and
A41:    Y<>M and
A42:    a,b // a1,b1 and
A43:    a,c // a1,c1 and
A44:    LIN b,c,d and
A45:    LIN b1,c1,d;
        LIN a,a1,d or b,c // b1,c1
        proof
          assume that
A46:      not LIN a,a1,d and
A47:      not b,c // b1,c1;
A48:      c <>c1 & a<>a1 & b<>b1
          proof
A49:        now
A50:          not LIN o,a,c
              proof
                assume LIN o,a,c;
                then c in Y by A25,A28,A31,A37,AFF_1:25;
                then
A51:            o,c // Y by A25,A37,AFF_1:52;
                o,c // M by A27,A35,A39,AFF_1:52;
                hence contradiction by A25,A27,A30,A41,A51,AFF_1:45,53;
              end;
              o,c // o,c1 by A27,A35,A36,A39,AFF_1:39,41;
              then
A52:          LIN o,c,c1 by AFF_1:def 1;
A53:          not LIN o,b,a
              proof
                assume LIN o,b,a;
                then a in Z by A26,A29,A33,A38,AFF_1:25;
                then
A54:            o,a // Z by A26,A38,AFF_1:52;
                o,a // Y by A25,A31,A37,AFF_1:52;
                hence contradiction by A25,A26,A28,A40,A54,AFF_1:45,53;
              end;
              o,a // o,a1 by A25,A31,A32,A37,AFF_1:39,41;
              then
A55:          LIN o,a,a1 by AFF_1:def 1;
              assume
A56:          b=b1;
              then b,a // b,a1 by A42,AFF_1:4;
              then a=a1 by A53,A55,AFF_1:14;
              then c =c1 by A43,A50,A52,AFF_1:14;
              hence contradiction by A47,A56,AFF_1:2;
            end;
A57:        now
A58:          not LIN o,a,b
              proof
                assume LIN o,a,b;
                then b in Y by A25,A28,A31,A37,AFF_1:25;
                then
A59:            o,b // Y by A25,A37,AFF_1:52;
                o,b // Z by A26,A33,A38,AFF_1:52;
                hence contradiction by A25,A26,A29,A40,A59,AFF_1:45,53;
              end;
              o,b // o,b1 by A26,A33,A34,A38,AFF_1:39,41;
              then
A60:          LIN o,b,b1 by AFF_1:def 1;
A61:          not LIN o,c,a
              proof
                assume LIN o,c,a;
                then a in M by A27,A30,A35,A39,AFF_1:25;
                then
A62:            o,a // M by A27,A39,AFF_1:52;
                o,a // Y by A25,A31,A37,AFF_1:52;
                hence contradiction by A25,A27,A28,A41,A62,AFF_1:45,53;
              end;
              o,a // o,a1 by A25,A31,A32,A37,AFF_1:39,41;
              then
A63:          LIN o,a,a1 by AFF_1:def 1;
              assume
A64:          c =c1;
              then c,a // c,a1 by A43,AFF_1:4;
              then a=a1 by A61,A63,AFF_1:14;
              then b=b1 by A42,A58,A60,AFF_1:14;
              hence contradiction by A47,A64,AFF_1:2;
            end;
A65:        now
A66:          not LIN o,a,c
              proof
                assume LIN o,a,c;
                then c in Y by A25,A28,A31,A37,AFF_1:25;
                then
A67:            o,c // Y by A25,A37,AFF_1:52;
                o,c // M by A27,A35,A39,AFF_1:52;
                hence contradiction by A25,A27,A30,A41,A67,AFF_1:45,53;
              end;
A68:          not LIN o,a,b
              proof
                assume LIN o,a,b;
                then b in Y by A25,A28,A31,A37,AFF_1:25;
                then
A69:            o,b // Y by A25,A37,AFF_1:52;
                o,b // Z by A26,A33,A38,AFF_1:52;
                hence contradiction by A25,A26,A29,A40,A69,AFF_1:45,53;
              end;
              assume
A70:          a=a1;
              o,c // o,c1 by A27,A35,A36,A39,AFF_1:39,41;
              then LIN o,c,c1 by AFF_1:def 1;
              then
A71:          c =c1 by A43,A70,A66,AFF_1:14;
              o,b // o,b1 by A26,A33,A34,A38,AFF_1:39,41;
              then LIN o,b,b1 by AFF_1:def 1;
              then b=b1 by A42,A70,A68,AFF_1:14;
              hence contradiction by A47,A71,AFF_1:2;
            end;
            assume not thesis;
            hence contradiction by A57,A65,A49;
          end;
          now
            assume
A72:        b<>c;
A73:        now
              assume
A74:          b1<>o;
A75:          a1<>b1 & a1<>c1
              proof
A76:            now
A77:              o<>a1
                  proof
A78:                not LIN a,b,o
                    proof
                      assume LIN a,b,o;
                      then LIN o,a,b by AFF_1:6;
                      then b in Y by A25,A28,A31,A37,AFF_1:25;
                      then
A79:                  o,b // Y by A25,A37,AFF_1:52;
                      o,b // Z by A26,A33,A38,AFF_1:52;
                      hence contradiction by A25,A26,A29,A40,A79,AFF_1:45,53;
                    end;
A80:                o,b // o,b1 by A26,A33,A34,A38,AFF_1:39,41;
                    assume o=a1;
                    then a,b // o,b by A42,A74,A80,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 A29,A80,AFF_1:5;
                    then LIN o,a,b1 by AFF_1:def 1;
                    then
A81:                LIN a,o,b1 by AFF_1:6;
                    b,o // b,b1 by A26,A33,A34,A38,AFF_1:39,41;
                    hence contradiction by A74,A78,A81,AFF_1:14;
                  end;
                  assume a1=c1;
                  then
A82:              o,a1 // M by A27,A36,A39,AFF_1:52;
                  o,a1 // Y by A25,A32,A37,AFF_1:52;
                  hence contradiction by A25,A27,A41,A82,A77,AFF_1:45,53;
                end;
A83:            now
                  assume a1=b1;
                  then
A84:              o,b1 // Y by A25,A32,A37,AFF_1:52;
                  o,b1 // Z by A26,A34,A38,AFF_1:52;
                  hence contradiction by A25,A26,A40,A74,A84,AFF_1:45,53;
                end;
                assume not thesis;
                hence contradiction by A83,A76;
              end;
              LIN o,o,a by AFF_1:7;
              then consider C such that
A85:          C is being_line and
A86:          o in C and
              o in C and
A87:          a in C by AFF_1:21;
A88:          ex d2 st d2 in C & a,c // d,d2
              proof
                consider e2 such that
A89:            a,c // d,e2 and
A90:            d<>e2 by DIRAF:40;
                consider e1 such that
A91:            o,a // o,e1 and
A92:            o<>e1 by DIRAF:40;
                LIN o,a,e1 by A91,AFF_1:def 1;
                then
A93:            e1 in C by A28,A85,A86,A87,AFF_1:25;
                not o,e1 // d,e2
                proof
                  assume o,e1 // d,e2;
                  then o,a // d,e2 by A91,A92,AFF_1:5;
                  then o,a // a,c by A89,A90,AFF_1:5;
                  then a,o // a,c by AFF_1:4;
                  then LIN a,o,c by AFF_1:def 1;
                  then c in Y by A25,A28,A31,A37,AFF_1:25;
                  then
A94:              o, c // Y by A25,A37,AFF_1:52;
                  o,c // M by A27,A35,A39,AFF_1:52;
                  hence contradiction by A25,A27,A30,A41,A94,AFF_1:45,53;
                end;
                then consider d2 such that
A95:            LIN o,e1,d2 and
A96:            LIN d,e2,d2 by AFF_1:60;
                take d2;
                d,e2 // d,d2 by A96,AFF_1:def 1;
                hence thesis by A85,A86,A92,A89,A90,A95,A93,AFF_1:5,25;
              end;
A97:          ex d1 st d1 in C & c,c1 // d,d1
              proof
                consider e2 such that
A98:            c,c1 // d,e2 and
A99:            d<>e2 by DIRAF:40;
                consider e1 such that
A100:           o,a // o,e1 and
A101:           o<>e1 by DIRAF:40;
                LIN o,a,e1 by A100,AFF_1:def 1;
                then
A102:           e1 in C by A28,A85,A86,A87,AFF_1:25;
                not o,e1 // d,e2
                proof
                  assume o,e1 // d,e2;
                  then o,a // d,e2 by A100,A101,AFF_1:5;
                  then
A103:             o,a // c,c1 by A98,A99,AFF_1:5;
                  c,c1 // o,c by A27,A35,A36,A39,AFF_1:39,41;
                  then o,a // o,c by A48,A103,AFF_1:5;
                  then LIN o,a,c by AFF_1:def 1;
                  then c in Y by A25,A28,A31,A37,AFF_1:25;
                  then
A104:             o,c // Y by A25,A37,AFF_1:52;
                  o,c // M by A27,A35,A39,AFF_1:52;
                  hence contradiction by A25,A27,A30,A41,A104,AFF_1:45,53;
                end;
                then consider d1 such that
A105:           LIN o,e1,d1 and
A106:           LIN d,e2,d1 by AFF_1:60;
                take d1;
                d,e2 // d,d1 by A106,AFF_1:def 1;
                hence thesis by A85,A86,A101,A98,A99,A105,A102,AFF_1:5,25;
              end;
              consider d2 such that
A107:         d2 in C and
A108:         a,c // d,d2 by A88;
              consider d1 such that
A109:         d1 in C and
A110:         c,c1 // d,d1 by A97;
              c,c1 // c,o by A27,A35,A36,A39,AFF_1:39,41;
              then
A111:         c,o // d,d1 by A48,A110,AFF_1:5;
              o,a // o,a1 by A25,A31,A32,A37,AFF_1:39,41;
              then LIN o,a,a1 by AFF_1:def 1;
              then
A112:         a1 in C by A28,A85,A86,A87,AFF_1:25;
              LIN b,b,c by AFF_1:7;
              then consider A such that
A113:         A is being_line and
A114:         b in A and
              b in A and
A115:         c in A by AFF_1:21;
A116:         d in A by A44,A72,A113,A114,A115,AFF_1:25;
A117:         b<>c by A47,AFF_1:3;
A118:         ex d3 st d3 in A & o,b // d1,d3
              proof
                consider e2 such that
A119:           o,b // d1,e2 and
A120:           d1<>e2 by DIRAF:40;
                consider e1 such that
A121:           b,c // b,e1 and
A122:           b<>e1 by DIRAF:40;
                LIN b,c,e1 by A121,AFF_1:def 1;
                then
A123:           e1 in A by A117,A113,A114,A115,AFF_1:25;
                not b,e1 // d1,e2
                proof
                  assume b,e1 // d1,e2;
                  then b,c // d1,e2 by A121,A122,AFF_1:5;
                  then
A124:             b,c // o,b by A119,A120,AFF_1:5;
                  then b,c // b,o by AFF_1:4;
                  then LIN b,c,o by AFF_1:def 1;
                  then LIN o,b,c by AFF_1:6;
                  then
A125:             o, b // o,c by AFF_1:def 1;
A126:             o,b // o,b1 by A26,A33,A34,A38,AFF_1:39,41;
                  o,c // o,c1 by A27,A35,A36,A39,AFF_1:39,41;
                  then o,b // o,c1 by A30,A125,AFF_1:5;
                  then o,b1 // o,c1 by A29,A126,AFF_1:5;
                  then LIN o, b1,c1 by AFF_1:def 1;
                  then LIN b1,c1,o by AFF_1:6;
                  then b1,c1 // b1,o by AFF_1:def 1;
                  then b1,c1 // o,b1 by AFF_1:4;
                  then b1,c1 // o,b by A74,A126,AFF_1:5;
                  hence contradiction by A29,A47,A124,AFF_1:5;
                end;
                then consider d3 such that
A127:           LIN b,e1,d3 and
A128:           LIN d1,e2,d3 by AFF_1:60;
                take d3;
                d1,e2 // d1,d3 by A128,AFF_1:def 1;
                hence thesis by A113,A114,A122,A119,A120,A127,A123,AFF_1:5,25;
              end;
A129:         not o in A & not a in A & not d1 in A & not d2 in A
              proof
A130:           now
A131:             a<>b
                  proof
                    assume a=b;
                    then
A132:               o,b // Y by A25,A31,A37,AFF_1:52;
                    o,b // Z by A26,A33,A38,AFF_1:52;
                    hence contradiction by A25,A26,A29,A40,A132,AFF_1:45,53;
                  end;
A133:             a1<>b1
                  proof
                    assume a1=b1;
                    then
A134:               o,b1 // Y by A25,A32,A37,AFF_1:52;
                    o,b1 // Z by A26,A34,A38,AFF_1:52;
                    hence contradiction by A25,A26,A40,A74,A134,AFF_1:45,53;
                  end;
                  assume a in A;
                  then
A135:             a,b // a,c by A113,A114,A115,AFF_1:39,41;
                  a<>c
                  proof
                    assume a=c;
                    then
A136:               o,c // Y by A25,A31,A37,AFF_1:52;
                    o,c // M by A27,A35,A39,AFF_1:52;
                    hence contradiction by A25,A27,A30,A41,A136,AFF_1:45,53;
                  end;
                  then a,b // a1,c1 by A43,A135,AFF_1:5;
                  then a1,b1 // a1,c1 by A42,A131,AFF_1:5;
                  then LIN a1,b1,c1 by AFF_1:def 1;
                  then LIN b1,c1,a1 by AFF_1:6;
                  then b1,c1 // b1,a1 by AFF_1:def 1;
                  then
A137:             a1,b1 // b1,c1 by AFF_1:4;
                  LIN a,b,c by A135,AFF_1:def 1;
                  then LIN b,c,a by AFF_1:6;
                  then b, c // b,a by AFF_1:def 1;
                  then b,c // a,b by AFF_1:4;
                  then b,c // a1,b1 by A42,A131,AFF_1:5;
                  hence contradiction by A47,A137,A133,AFF_1:5;
                end;
A138:           now
A139:             d<>d2
                  proof
                    o,a // o,a1 by A25,A31,A32,A37,AFF_1:39,41;
                    then LIN o,a,a1 by AFF_1:def 1;
                    then
A140:               a1 in C by A28,A85,A86,A87,AFF_1:25;
                    assume d=d2;
                    then a,a1 // a,d by A85,A87,A107,A140,AFF_1:39,41;
                    hence contradiction by A46,AFF_1:def 1;
                  end;
                  assume
A141:             d2 in A;
A142:             d,d2 // c,a by A108,AFF_1:4;
                  d in A by A44,A117,A113,A114,A115,AFF_1:25;
                  hence contradiction by A113,A115,A130,A141,A139,A142,AFF_1:48
;
                end;
A143:           now
                  assume o in A;
                  then
A144:             o,b // o,c by A113,A114,A115,AFF_1:39,41;
A145:             o,b // o,b1 by A26,A33,A34,A38,AFF_1:39,41;
                  o,c // o,c1 by A27,A35,A36,A39,AFF_1:39,41;
                  then o,b // o,c1 by A30,A144,AFF_1:5;
                  then o,b1 // o,c1 by A29,A145,AFF_1:5;
                  then LIN o,b1,c1 by AFF_1:def 1;
                  then LIN b1,c1,o by AFF_1:6;
                  then b1,c1 // b1,o by AFF_1:def 1;
                  then b1,c1 // o,b1 by AFF_1:4;
                  then o,b // b1,c1 by A74,A145,AFF_1:5;
                  then
A146:             b,o // b1,c1 by AFF_1:4;
                  LIN o,b,c by A144,AFF_1:def 1;
                  then LIN b,c,o by AFF_1:6;
                  then b,c // b,o by AFF_1:def 1;
                  hence contradiction by A29,A47,A146,AFF_1:5;
                end;
A147:           now
A148:             d<>d1
                  proof
                    o,a // o,a1 by A25,A31,A32,A37,AFF_1:39,41;
                    then LIN o,a,a1 by AFF_1:def 1;
                    then
A149:               a1 in C by A28,A85,A86,A87,AFF_1:25;
                    assume d=d1;
                    then a,a1 // a,d by A85,A87,A109,A149,AFF_1:39,41;
                    hence contradiction by A46,AFF_1:def 1;
                  end;
                  assume
A150:             d1 in A;
                  c,c1 // c,o by A27,A35,A36,A39,AFF_1:39,41;
                  then
A151:             LIN c,c1,o by AFF_1:def 1;
A152:             d,d1 // c,c1 by A110,AFF_1:4;
                  d in A by A44,A117,A113,A114,A115,AFF_1:25;
                  then c1 in A by A113,A115,A150,A148,A152,AFF_1:48;
                  hence contradiction by A48,A113,A115,A143,A151,AFF_1:25;
                end;
                assume not thesis;
                hence contradiction by A143,A130,A147,A138;
              end;
              LIN b1,b1,c1 by AFF_1:7;
              then consider K such that
A153:         K is being_line and
A154:         b1 in K and
              b1 in K and
A155:         c1 in K by AFF_1:21;
              b1<>c1 by A47,AFF_1:3;
              then
A156:         d in K by A45,A153,A154,A155,AFF_1:25;
              consider d3 such that
A157:         d3 in A and
A158:         o,b // d1,d3 by A118;
A159:         b1<>c1 by A47,AFF_1:3;
              ex d4 st d4 in K & d1,d3 // d1,d4
              proof
                consider e2 such that
A160:           d1,d3 // d1,e2 and
A161:           d1<>e2 by DIRAF:40;
                consider e1 such that
A162:           b1,c1 // b1,e1 and
A163:           b1<>e1 by DIRAF:40;
                LIN b1,c1,e1 by A162,AFF_1:def 1;
                then
A164:           e1 in K by A159,A153,A154,A155,AFF_1:25;
                not b1,e1 // d1,e2
                proof
A165:             o<>c1
                  proof
A166:               not LIN a,c,o
                    proof
                      assume LIN a,c,o;
                      then LIN o,a,c by AFF_1:6;
                      then c in Y by A25,A28,A31,A37,AFF_1:25;
                      then
A167:                 o,c // Y by A25,A37,AFF_1:52;
                      o,c // M by A27,A35,A39,AFF_1:52;
                      hence contradiction by A25,A27,A30,A41,A167,AFF_1:45,53;
                    end;
                    assume
A168:               o=c1;
                    a1,o // a,a1 by A25,A31,A32,A37,AFF_1:39,41;
                    then a,c // a,a1 by A43,A75,A168,AFF_1:5;
                    then LIN a,c,a1 by AFF_1:def 1;
                    then LIN a1,c,a by AFF_1:6;
                    then
A169:               a1,c // a1,a by AFF_1:def 1;
                    a1,a // a1,o by A25,A31,A32,A37,AFF_1:39,41;
                    then a1,c // a1,o by A48,A169,AFF_1:5;
                    then LIN a1,c,o by AFF_1:def 1;
                    then LIN c,o,a1 by AFF_1:6;
                    then
A170:               c,o // c,a1 by AFF_1:def 1;
                    a,o // a,a1 by A25,A31,A32,A37,AFF_1:39,41;
                    then LIN a,o,a1 by AFF_1:def 1;
                    hence contradiction by A75,A168,A166,A170,AFF_1:14;
                  end;
A171:             d1<>d3
                  proof
A172:               d<>d1
                    proof
                      o,a // o,a1 by A25,A31,A32,A37,AFF_1:39,41;
                      then LIN o,a,a1 by AFF_1:def 1;
                      then
A173:                 a1 in C by A28,A85,A86,A87,AFF_1:25;
                      assume d=d1;
                      then a,a1 // a,d by A85,A87,A109,A173,AFF_1:39,41;
                      hence contradiction by A46,AFF_1:def 1;
                    end;
                    assume
A174:               d1=d3;
                    c,c1 // o,c by A27,A35,A36,A39,AFF_1:39,41;
                    then o,c // d,d1 by A48,A110,AFF_1:5;
                    then
A175:               d,d1 // c,o by AFF_1:4;
A176:               o,b // o,b1 by A26,A33,A34,A38,AFF_1:39,41;
                    d in A by A44,A117,A113,A114,A115,AFF_1:25;
                    then o in A by A113,A115,A157,A174,A172,A175,AFF_1:48;
                    then
A177:               o,b // o,c by A113,A114,A115,AFF_1:39,41;
                    o,c // o,c1 by A27,A35,A36,A39,AFF_1:39,41;
                    then o,b // o,c1 by A30,A177,AFF_1:5;
                    then o,b1 // o,c1 by A29,A176,AFF_1:5;
                    then LIN o,b1,c1 by AFF_1:def 1;
                    then LIN b1,c1,o by AFF_1:6;
                    then b1,c1 // b1,o by AFF_1:def 1;
                    then b1,c1 // o,b1 by AFF_1:4;
                    then
A178:               o,b // b1,c1 by A74,A176,AFF_1:5;
                    LIN o,b,c by A177,AFF_1:def 1;
                    then LIN b,c,o by AFF_1:6;
                    then b,c // b,o by AFF_1:def 1;
                    then b,c // o,b by AFF_1:4;
                    hence contradiction by A29,A47,A178,AFF_1:5;
                  end;
                  assume b1,e1 // d1,e2;
                  then b1,c1 // d1,e2 by A162,A163,AFF_1:5;
                  then
A179:             b1,c1 // d1,d3 by A160,A161,AFF_1:5;
A180:             o,b // o,b1 by A26,A33,A34,A38,AFF_1:39,41;
                  then o,b1 // d1,d3 by A29,A158,AFF_1:5;
                  then
A181:             o,b1 // b1,c1 by A179,A171,AFF_1:5;
                  then b1,o // b1,c1 by AFF_1:4;
                  then LIN b1,o,c1 by AFF_1:def 1;
                  then LIN o,b1,c1 by AFF_1:6;
                  then o,b1 // o,c1 by AFF_1:def 1;
                  then
A182:             o,b // o,c1 by A74,A180,AFF_1:5;
                  o,c // o,c1 by A27,A35,A36,A39,AFF_1:39,41;
                  then o,b // o,c by A182,A165,AFF_1:5;
                  then LIN o,b,c by AFF_1:def 1;
                  then LIN b,o,c by AFF_1:6;
                  then b,o // b,c by AFF_1:def 1;
                  then o,b // b,c by AFF_1:4;
                  then o,b1 // b,c by A29,A180,AFF_1:5;
                  hence contradiction by A47,A74,A181,AFF_1:5;
                end;
                then consider d4 such that
A183:           LIN b1,e1,d4 and
A184:           LIN d1,e2,d4 by AFF_1:60;
                take d4;
                d1,e2 // d1,d4 by A184,AFF_1:def 1;
                hence thesis by A153,A154,A163,A160,A161,A183,A164,AFF_1:5,25;
              end;
              then consider d4 such that
A185:         d4 in K and
A186:         d1,d3 // d1,d4;
A187:         not c in C & not b in C & not d in C & not d3 in C
              proof
A188:           now
                  assume b in C;
                  then o,a // o,b by A85,A86,A87,AFF_1:39,41;
                  then LIN o,a,b by AFF_1:def 1;
                  then b in Y by A25,A28,A31,A37,AFF_1:25;
                  then
A189:             o,b // Y by A25,A37,AFF_1:52;
                  o,b // Z by A26,A33,A38,AFF_1:52;
                  hence contradiction by A25,A26,A29,A40,A189,AFF_1:45,53;
                end;
A190:           now
A191:             d1<>d3
                  proof
A192:               d<>d1
                    proof
                      o,a // o,a1 by A25,A31,A32,A37,AFF_1:39,41;
                      then LIN o,a,a1 by AFF_1:def 1;
                      then
A193:                 a1 in C by A28,A85,A86,A87,AFF_1:25;
                      assume d=d1;
                      then a,a1 // a,d by A85,A87,A109,A193,AFF_1:39,41;
                      hence contradiction by A46,AFF_1:def 1;
                    end;
                    assume
A194:               d1=d3;
A195:               o,c // o,c1 by A27,A35,A36,A39,AFF_1:39,41;
A196:               d,d1 // c,c1 by A110,AFF_1:4;
                    d in A by A44,A117,A113,A114,A115,AFF_1:25;
                    then c1 in A by A113,A115,A157,A194,A192,A196,AFF_1:48;
                    then
A197:               c,c1 // c,b by A113,A114,A115,AFF_1:39,41;
A198:               o,b // o,b1 by A26,A33,A34,A38,AFF_1:39,41;
                    c,o // c,c1 by A27,A35,A36,A39,AFF_1:39,41;
                    then c,o // c,b by A48,A197,AFF_1:5;
                    then
A199:               LIN c,o,b by AFF_1:def 1;
                    then LIN o,b,c by AFF_1:6;
                    then o,b // o,c by AFF_1:def 1;
                    then o,b1 // o,c by A29,A198,AFF_1:5;
                    then o,b1 // o,c1 by A30,A195,AFF_1:5;
                    then LIN o,b1,c1 by AFF_1:def 1;
                    then LIN b1,c1,o by AFF_1:6;
                    then b1,c1 // b1,o by AFF_1:def 1;
                    then b1,c1 // o,b1 by AFF_1:4;
                    then
A200:               o,b // b1,c1 by A74,A198,AFF_1:5;
                    LIN b,c,o by A199,AFF_1:6;
                    then b,c // b,o by AFF_1:def 1;
                    then b,c // o,b by AFF_1:4;
                    hence contradiction by A29,A47,A200,AFF_1:5;
                  end;
                  assume
A201:             d3 in C;
                  d1,d3 // o,b by A158,AFF_1:4;
                  hence contradiction by A85,A86,A109,A188,A201,A191,AFF_1:48;
                end;
A202:           now
                  assume c in C;
                  then o,a // o,c by A85,A86,A87,AFF_1:39,41;
                  then LIN o,a,c by AFF_1:def 1;
                  then c in Y by A25,A28,A31,A37,AFF_1:25;
                  then
A203:             o,c // Y by A25,A37,AFF_1:52;
                  o,c // M by A27,A35,A39,AFF_1:52;
                  hence contradiction by A25,A27,A30,A41,A203,AFF_1:45,53;
                end;
A204:           now
                  assume d in C;
                  then
A205:             o,a // a,d by A85,A86,A87,AFF_1:39,41;
                  o,a // a,a1 by A25,A31,A32,A37,AFF_1:39,41;
                  then a,a1 // a,d by A28,A205,AFF_1:5;
                  hence contradiction by A46,AFF_1:def 1;
                end;
                assume not thesis;
                hence contradiction by A202,A188,A204,A190;
              end;
A206:         d4<>d3
              proof
                assume
A207:           d4=d3;
                d=d3
                proof
                  d in A by A44,A117,A113,A114,A115,AFF_1:25;
                  then
A208:             b,c // d,d3 by A113,A114,A115,A157,AFF_1:39,41;
                  d in K by A45,A159,A153,A154,A155,AFF_1:25;
                  then
A209:             b1,c1 // d,d3 by A153,A154,A155,A185,A207,AFF_1:39,41;
                  assume d<>d3;
                  hence contradiction by A47,A209,A208,AFF_1:5;
                end;
                then
A210:           o,b // d,d1 by A158,AFF_1:4;
                o,c // c,c1 by A27,A35,A36,A39,AFF_1:39,41;
                then o,c // d,d1 by A48,A110,AFF_1:5;
                then
A211:           o,b // o,c by A109,A187,A210,AFF_1:5;
A212:           o,c // o,c1 by A27,A35,A36,A39,AFF_1:39,41;
A213:           o,b // o,b1 by A26,A33,A34,A38,AFF_1:39,41;
                then o,b1 // o,c by A29,A211,AFF_1:5;
                then o,b1 // o,c1 by A30,A212,AFF_1:5;
                then LIN o,b1,c1 by AFF_1:def 1;
                then LIN b1,c1,o by AFF_1:6;
                then b1,c1 // b1,o by AFF_1:def 1;
                then b1,c1 // o,b1 by AFF_1:4;
                then
A214:           o,b // b1,c1 by A74,A213,AFF_1:5;
                LIN o,b,c by A211,AFF_1:def 1;
                then LIN b,c,o by AFF_1:6;
                then b,c // b,o by AFF_1:def 1;
                then b,c // o,b by AFF_1:4;
                hence contradiction by A29,A47,A214,AFF_1:5;
              end;
A215:         a<>b & a<>c
              proof
A216:           now
                  assume a=c;
                  then
A217:             o,a // M by A27,A35,A39,AFF_1:52;
                  o,a // Y by A25,A31,A37,AFF_1:52;
                  hence contradiction by A25,A27,A28,A41,A217,AFF_1:45,53;
                end;
A218:           now
                  assume a=b;
                  then
A219:             o,a // Z by A26,A33,A38,AFF_1:52;
                  o,a // Y by A25,A31,A37,AFF_1:52;
                  hence contradiction by A25,A26,A28,A40,A219,AFF_1:45,53;
                end;
                assume not thesis;
                hence contradiction by A218,A216;
              end;
A220:         not o in K & not a1 in K & not d1 in K & not d2 in K
              proof
A221:           now
A222:             o<>c1
                  proof
A223:               a1<>o
                    proof
A224:                 not LIN a,b,o
                      proof
                        assume LIN a,b,o;
                        then LIN o,a,b by AFF_1:6;
                        then b in Y by A25,A28,A31,A37,AFF_1:25;
                        then
A225:                   o,b // Y by A25,A37,AFF_1:52;
                        o,b // Z by A26,A33,A38,AFF_1:52;
                        hence contradiction by A25,A26,A29,A40,A225,AFF_1:45,53
;
                      end;
A226:                 o,b // o,b1 by A26,A33,A34,A38,AFF_1:39,41;
                      assume a1=o;
                      then a,b // o,b by A42,A74,A226,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 A29,A226,AFF_1:5;
                      then LIN o,a,b1 by AFF_1:def 1;
                      then
A227:                 LIN a,o,b1 by AFF_1:6;
                      b,o // b,b1 by A26,A33,A34,A38,AFF_1:39,41;
                      hence contradiction by A74,A224,A227,AFF_1:14;
                    end;
A228:               not LIN c,a,o
                    proof
                      assume LIN c,a,o;
                      then LIN o,a,c by AFF_1:6;
                      then c in Y by A25,A28,A31,A37,AFF_1:25;
                      then
A229:                 o,c // Y by A25,A37,AFF_1:52;
                      o,c // M by A27,A35,A39,AFF_1:52;
                      hence contradiction by A25,A27,A30,A41,A229,AFF_1:45,53;
                    end;
A230:               a1,o // o,a by A25,A31,A32,A37,AFF_1:39,41;
                    assume o=c1;
                    then a,c // o,a by A43,A223,A230,AFF_1:5;
                    then a,c // a,o by AFF_1:4;
                    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 // a1,o by A28,A230,AFF_1:5;
                    then o,c // o,a1 by AFF_1:4;
                    then LIN o,c,a1 by AFF_1:def 1;
                    then
A231:               LIN c,o,a1 by AFF_1:6;
                    a,o // a,a1 by A25,A31,A32,A37,AFF_1:39,41;
                    hence contradiction by A223,A228,A231,AFF_1:14;
                  end;
A232:             o,c // o,c1 by A27,A35,A36,A39,AFF_1:39,41;
                  assume o in K;
                  then
A233:             o,b1 // o,c1 by A153,A154,A155,AFF_1:39,41;
A234:             o,b1 // o,b by A26,A33,A34,A38,AFF_1:39,41;
                  then o,b // o,c1 by A74,A233,AFF_1:5;
                  then o,b // o,c by A222,A232,AFF_1:5;
                  then LIN o,b,c by AFF_1:def 1;
                  then LIN b,c,o by AFF_1:6;
                  then b,c // b,o by AFF_1:def 1;
                  then b,c // o,b by AFF_1:4;
                  then b,c // o,b1 by A29,A234,AFF_1:5;
                  then
A235:             b,c // b1,o by AFF_1:4;
                  LIN o,b1,c1 by A233,AFF_1:def 1;
                  then LIN b1,c1,o by AFF_1:6;
                  then b1,c1 // b1,o by AFF_1:def 1;
                  hence contradiction by A47,A74,A235,AFF_1:5;
                end;
A236:           now
A237:             d<>d1
                  proof
                    assume d=d1;
                    then
A238:               o,a // a,d by A85,A86,A87,A109,AFF_1:39,41;
                    o,a // a,a1 by A25,A31,A32,A37,AFF_1:39,41;
                    then a,a1 // a,d by A28,A238,AFF_1:5;
                    hence contradiction by A46,AFF_1:def 1;
                  end;
                  assume
A239:             d1 in K;
                  c,c1 // c,o by A27,A35,A36,A39,AFF_1:39,41;
                  then
A240:             LIN c,c1,o by AFF_1:def 1;
A241:             d,d1 // c1,c by A110,AFF_1:4;
                  d in K by A45,A159,A153,A154,A155,AFF_1:25;
                  then c in K by A153,A155,A239,A237,A241,AFF_1:48;
                  hence contradiction by A48,A153,A155,A221,A240,AFF_1:25;
                end;
A242:           now
                  assume a1 in K;
                  then
A243:             a1,b1 // a1,c1 by A153,A154,A155,AFF_1:39,41;
                  then a,b // a1,c1 by A42,A75,AFF_1:5;
                  then a,b // a,c by A43,A75,AFF_1:5;
                  then LIN a,b,c by AFF_1:def 1;
                  then LIN b,c,a by AFF_1:6;
                  then b,c // b,a by AFF_1:def 1;
                  then b,c // a,b by AFF_1:4;
                  then
A244:             b,c // a1,b1 by A42,A215,AFF_1:5;
                  LIN a1,b1,c1 by A243,AFF_1:def 1;
                  then LIN b1,c1,a1 by AFF_1:6;
                  then b1,c1 // b1,a1 by AFF_1:def 1;
                  then b1,c1 // a1,b1 by AFF_1:4;
                  hence contradiction by A47,A75,A244,AFF_1:5;
                end;
A245:           now
A246:             d<>d2
                  proof
                    assume d=d2;
                    then
A247:               o,a // a,d by A85,A86,A87,A107,AFF_1:39,41;
                    o,a // a,a1 by A25,A31,A32,A37,AFF_1:39,41;
                    then a,a1 // a,d by A28,A247,AFF_1:5;
                    hence contradiction by A46,AFF_1:def 1;
                  end;
                  assume
A248:             d2 in K;
                  d,d2 // a1,c1 by A43,A215,A108,AFF_1:5;
                  then
A249:             d,d2 // c1,a1 by AFF_1:4;
                  d in K by A45,A159,A153,A154,A155,AFF_1:25;
                  hence contradiction by A153,A155,A242,A248,A246,A249,AFF_1:48
;
                end;
                assume not thesis;
                hence contradiction by A221,A242,A236,A245;
              end;
A250:         not c1 in C & not b1 in C & not d in C & not d4 in C
              proof
A251:           now
                  assume
A252:             c1 in C;
                  o,c1 // c1,c by A27,A35,A36,A39,AFF_1:39,41;
                  hence contradiction by A155,A85,A86,A187,A220,A252,AFF_1:48;
                end;
A253:           now
                  assume
A254:             d4 in C;
                  d1,d4 // d1,d3 by A186,AFF_1:4;
                  hence contradiction by A85,A109,A185,A187,A220,A254,AFF_1:48;
                end;
A255:           now
                  assume
A256:             b1 in C;
                  o,b1 // o,b by A26,A33,A34,A38,AFF_1:39,41;
                  hence contradiction by A74,A85,A86,A187,A256,AFF_1:48;
                end;
                assume not thesis;
                hence contradiction by A187,A251,A255,A253;
              end;
              a<>c
              proof
                assume a=c;
                then
A257:           o,a // M by A27,A35,A39,AFF_1:52;
                o,a // Y by A25,A31,A37,AFF_1:52;
                hence contradiction by A25,A27,A28,A41,A257,AFF_1:45,53;
              end;
              then a1,c1 // d,d2 by A43,A108,AFF_1:5;
              then
A258:         a1,c1 // d2,d by AFF_1:4;
              c1,o // c,c1 by A27,A35,A36,A39,AFF_1:39,41;
              then
A259:         c1,o // d,d1 by A48,A110,AFF_1:5;
A260:         d1<>d2
              proof
                assume d1=d2;
                then a,c // c,c1 by A109,A110,A108,A187,AFF_1:5;
                then c,c1 // c,a by AFF_1:4;
                then LIN c,c1,a by AFF_1:def 1;
                then a in M by A35,A36,A39,A48,AFF_1:25;
                then
A261:           o,a // M by A27,A39,AFF_1:52;
                o,a // Y by A25,A31,A37,AFF_1:52;
                hence contradiction by A25,A27,A28,A41,A261,AFF_1:45,53;
              end;
A262:         not LIN d4,d2,d1
              proof
                assume LIN d4,d2,d1;
                then LIN d1,d2,d4 by AFF_1:6;
                hence contradiction by A85,A109,A107,A250,A260,AFF_1:25;
              end;
A263:         o,b // o,b1 by A26,A33,A34,A38,AFF_1:51;
              a,c // d2,d by A108,AFF_1:4;
              then a,b // d2,d3 by A24,A113,A114,A115,A85,A86,A87,A109,A107
,A157,A158,A116,A129,A187,A111;
              then
A264:         d2,d3 // a1,b1 by A42,A215,AFF_1:5;
              o,b // d1,d4 by A109,A158,A186,A187,AFF_1:5;
              then o,b1 // d1,d4 by A29,A263,AFF_1:5;
              then a1,b1 // d2,d4 by A24,A153,A154,A155,A85,A86,A109,A107,A185
,A112,A156,A220,A250,A258,A259;
              then d2,d3 // d2,d4 by A75,A264,AFF_1:5;
              then LIN d2,d3,d4 by AFF_1:def 1;
              then LIN d4,d3,d2 by AFF_1:6;
              then
A265:         d4,d3 // d4,d2 by AFF_1:def 1;
              LIN d1,d3,d4 by A186,AFF_1:def 1;
              then LIN d4,d3,d1 by AFF_1:6;
              then d4,d3 // d4,d1 by AFF_1:def 1;
              then d4,d2 // d4,d1 by A206,A265,AFF_1:5;
              hence contradiction by A262,AFF_1:def 1;
            end;
            now
              assume
A266:         b1=o;
A267:         o=a1
              proof
A268:           not LIN b,a,o
                proof
                  assume LIN b,a,o;
                  then LIN o,a,b by AFF_1:6;
                  then b in Y by A25,A28,A31,A37,AFF_1:25;
                  then
A269:             o,b // Y by A25,A37,AFF_1:52;
                  o,b // Z by A26,A33,A38,AFF_1:52;
                  hence contradiction by A25,A26,A29,A40,A269,AFF_1:45,53;
                end;
A270:           a1<>a
                proof
                  assume a1=a;
                  then LIN a,b,o by A42,A266,AFF_1:def 1;
                  then LIN o,a,b by AFF_1:6;
                  then b in Y by A25,A28,A31,A37,AFF_1:25;
                  then
A271:             o,b // Y by A25,A37,AFF_1:52;
                  o,b // Z by A26,A33,A38,AFF_1:52;
                  hence contradiction by A25,A26,A29,A40,A271,AFF_1:45,53;
                end;
                assume
A272:           o<>a1;
                a1,o // a,a1 by A25,A31,A32,A37,AFF_1:51;
                then a,b // a,a1 by A42,A266,A272,AFF_1:5;
                then LIN a,b,a1 by AFF_1:def 1;
                then LIN a1,b,a by AFF_1:6;
                then
A273:           a1,b // a1,a by AFF_1:def 1;
                a1,a // a1,o by A25,A31,A32,A37,AFF_1:51;
                then a1,b // a1,o by A273,A270,AFF_1:5;
                then LIN a1,b,o by AFF_1:def 1;
                then LIN b,o,a1 by AFF_1:6;
                hence contradiction by A25,A31,A32,A37,A272,A268,AFF_1:14,51;
              end;
              o=c1
              proof
A274:           not LIN a,c,o
                proof
                  assume LIN a,c,o;
                  then LIN o,a,c by AFF_1:6;
                  then c in Y by A25,A28,A31,A37,AFF_1:25;
                  then
A275:             o,c // Y by A25,A37,AFF_1:52;
                  o,c // M by A27,A35,A39,AFF_1:52;
                  hence contradiction by A25,A27,A30,A41,A275,AFF_1:45,53;
                end;
                assume
A276:           o<>c1;
A277:           o,c1 // o,c by A27,A35,A36,A39,AFF_1:51;
                then o,c // a,c by A43,A267,A276,AFF_1:5;
                then c,o // c,a by AFF_1:4;
                then LIN c,o,a 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 A30,A277,AFF_1:5;
                then LIN o,a,c1 by AFF_1:def 1;
                then LIN a,o,c1 by AFF_1:6;
                hence contradiction by A27,A35,A36,A39,A276,A274,AFF_1:14,51;
              end;
              hence contradiction by A47,A266,AFF_1:3;
            end;
            hence contradiction by A73;
          end;
          hence contradiction by A47,AFF_1:3;
        end;
        hence LIN a,a1,d or b,c // b1,c1;
      end;
A278: now
A279:   M // M by A17,AFF_1:41;
        consider d such that
A280:   LIN b,c,d and
A281:   LIN b1,c1,d by A22,AFF_1:60;
A282:   LIN a,a1,d by A2,A3,A4,A5,A6,A7,A8,A9,A10,A11,A12,A13,A14,A15,A16,A17
,A18,A19,A20,A21,A22,A23,A280,A281;
A283:   not a in Z & not a in M
        proof
A284:     now
            assume a in M;
            then
A285:       o,a // M by A5,A17,AFF_1:52;
            o,a // Y by A3,A9,A15,AFF_1:52;
            hence contradiction by A3,A5,A6,A19,A285,AFF_1:45,53;
          end;
A286:     now
            assume a in Z;
            then
A287:       o,a // Z by A4,A16,AFF_1:52;
            o,a // Y by A3,A9,A15,AFF_1:52;
            hence contradiction by A3,A4,A6,A18,A287,AFF_1:45,53;
          end;
          assume not thesis;
          hence contradiction by A286,A284;
        end;
A288:   ex d1 st a,b // d,d1 & d1 in Z
        proof
          consider e1 such that
A289:     o,b // o,e1 and
A290:     o<>e1 by DIRAF:40;
          consider e2 such that
A291:     a,b // d,e2 and
A292:     d<>e2 by DIRAF:40;
          not o,e1 // d,e2
          proof
            assume o,e1 // d,e2;
            then o,b // d,e2 by A289,A290,AFF_1:5;
            then o,b // a,b by A291,A292,AFF_1:5;
            then b,o // b,a by AFF_1:4;
            then LIN b,o,a by AFF_1:def 1;
            hence contradiction by A4,A7,A11,A16,A283,AFF_1:25;
          end;
          then consider d1 such that
A293:     LIN o,e1,d1 and
A294:     LIN d,e2,d1 by AFF_1:60;
          o,e1 // o,d1 by A293,AFF_1:def 1;
          then o,b // o,d1 by A289,A290,AFF_1:5;
          then
A295:     LIN o,b,d1 by AFF_1:def 1;
          take d1;
          d,e2 // d,d1 by A294,AFF_1:def 1;
          hence thesis by A4,A7,A11,A16,A291,A292,A295,AFF_1:5,25;
        end;
A296:   Z // Z by A16,AFF_1:41;
A297:   a<>a1 & b<>b1 & c <>c1
        proof
A298:     now
A299:       not LIN a,o,b
            proof
              assume LIN a,o,b;
              then LIN o,b,a by AFF_1:6;
              hence contradiction by A4,A7,A11,A16,A283,AFF_1:25;
            end;
A300:       not LIN a,o,c
            proof
              assume LIN a,o,c;
              then LIN o,c,a by AFF_1:6;
              hence contradiction by A5,A8,A13,A17,A283,AFF_1:25;
            end;
            assume
A301:       a=a1;
            then LIN a,c,c1 by A21,AFF_1:def 1;
            then
A302:       c =c1 by A5,A13,A14,A279,A300,AFF_1:14,39;
            LIN a,b,b1 by A20,A301,AFF_1:def 1;
            then b=b1 by A4,A11,A12,A296,A299,AFF_1:14,39;
            hence contradiction by A22,A302,AFF_1:2;
          end;
A303:     now
            assume b=b1;
            then b,a // b,a1 by A20,AFF_1:4;
            then
A304:       LIN b,a,a1 by AFF_1:def 1;
            o,a // o,a1 by A3,A9,A10,A15,AFF_1:39,41;
            hence contradiction by A4,A7,A11,A16,A283,A298,A304,AFF_1:14,25;
          end;
A305:     now
            assume c =c1;
            then c,a // c,a1 by A21,AFF_1:4;
            then
A306:       LIN c,a,a1 by AFF_1:def 1;
            o,a // o,a1 by A3,A9,A10,A15,AFF_1:39,41;
            hence contradiction by A5,A8,A13,A17,A283,A298,A306,AFF_1:14,25;
          end;
          assume not thesis;
          hence contradiction by A298,A303,A305;
        end;
A307:   a1<>o & b1<>o & c1<>o
        proof
A308:     now
            assume
A309:       a1=o;
A310:       o=c1
            proof
A311:         not LIN c,a,o
              proof
                assume LIN c,a,o;
                then LIN o,c,a by AFF_1:6;
                hence contradiction by A5,A8,A13,A17,A283,AFF_1:25;
              end;
              assume
A312:         o<>c1;
A313:         o,c1 // o,c by A5,A13,A14,A17,AFF_1:39,41;
              then a,c // o,c by A21,A309,A312,AFF_1:5;
              then c,o // c,a by AFF_1:4;
              then LIN c,o,a 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,c1 // o,a by A8,A313,AFF_1:5;
              then LIN o,c1,a by AFF_1:def 1;
              then LIN a,o,c1 by AFF_1:6;
              then
A314:         a,o // a,c1 by AFF_1:def 1;
              c,o // c,c1 by A5,A13,A14,A17,AFF_1:39,41;
              then LIN c,o,c1 by AFF_1:def 1;
              hence contradiction by A312,A311,A314,AFF_1:14;
            end;
            o=b1
            proof
A315:         not LIN b,a,o
              proof
                assume LIN b,a,o;
                then LIN o,b,a by AFF_1:6;
                hence contradiction by A4,A7,A11,A16,A283,AFF_1:25;
              end;
              assume
A316:         o<>b1;
A317:         o,b1 // o,b by A4,A11,A12,A16,AFF_1:39,41;
              then a,b // o,b by A20,A309,A316,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,A317,AFF_1:5;
              then LIN o,a,b1 by AFF_1:def 1;
              then LIN a,o,b1 by AFF_1:6;
              then
A318:         a,o // a,b1 by AFF_1:def 1;
              b,o // b,b1 by A4,A11,A12,A16,AFF_1:39,41;
              then LIN b,o,b1 by AFF_1:def 1;
              hence contradiction by A316,A315,A318,AFF_1:14;
            end;
            hence contradiction by A22,A310,AFF_1:3;
          end;
A319:     now
A320:       not LIN a,b,o
            proof
              assume LIN a,b,o;
              then LIN o,b,a by AFF_1:6;
              hence contradiction by A4,A7,A11,A16,A283,AFF_1:25;
            end;
A321:       a1,o // a,a1 by A3,A9,A10,A15,AFF_1:39,41;
            assume b1=o;
            then a,b // a, a1 by A20,A308,A321,AFF_1:5;
            then LIN a,b,a1 by AFF_1:def 1;
            then LIN a1,b,a by AFF_1:6;
            then a1,b // a1,a by AFF_1:def 1;
            then a1,b // a,a1 by AFF_1:4;
            then a1,b // a1,o by A297,A321,AFF_1:5;
            then LIN a1,b,o by AFF_1:def 1;
            then LIN b,o,a1 by AFF_1:6;
            then
A322:       b,o // b,a1 by AFF_1:def 1;
            a,o // a,a1 by A3,A9,A10,A15,AFF_1:39,41;
            then LIN a,o,a1 by AFF_1:def 1;
            hence contradiction by A308,A320,A322,AFF_1:14;
          end;
A323:     now
A324:       not LIN a,c,o
            proof
              assume LIN a,c,o;
              then LIN o,c,a by AFF_1:6;
              hence contradiction by A5,A8,A13,A17,A283,AFF_1:25;
            end;
A325:       a1,o // a,a1 by A3,A9,A10,A15,AFF_1:39,41;
            assume c1=o;
            then a,c // a, a1 by A21,A308,A325,AFF_1:5;
            then LIN a,c,a1 by AFF_1:def 1;
            then LIN a1,a,c by AFF_1:6;
            then a1,a // a1,c by AFF_1:def 1;
            then a,a1 // a1,c by AFF_1:4;
            then a1,o // a1,c by A297,A325,AFF_1:5;
            then LIN a1,o,c by AFF_1:def 1;
            then LIN c,o,a1 by AFF_1:6;
            then
A326:       c,o // c,a1 by AFF_1:def 1;
            a,o // a,a1 by A3,A9,A10,A15,AFF_1:39,41;
            then LIN a,o,a1 by AFF_1:def 1;
            hence contradiction by A308,A324,A326,AFF_1:14;
          end;
          assume not thesis;
          hence contradiction by A308,A319,A323;
        end;
        ex d2 st a,c // d,d2 & d2 in M
        proof
          consider e1 such that
A327:     o,c // o,e1 and
A328:     o<>e1 by DIRAF:40;
          consider e2 such that
A329:     a,c // d,e2 and
A330:     d<>e2 by DIRAF:40;
          not o,e1 // d,e2
          proof
            assume o,e1 // d,e2;
            then o,c // d,e2 by A327,A328,AFF_1:5;
            then o,c // a,c by A329,A330,AFF_1:5;
            then c,o // c,a by AFF_1:4;
            then LIN c,o,a by AFF_1:def 1;
            hence contradiction by A5,A8,A13,A17,A283,AFF_1:25;
          end;
          then consider d2 such that
A331:     LIN o,e1,d2 and
A332:     LIN d,e2,d2 by AFF_1:60;
          o,e1 // o,d2 by A331,AFF_1:def 1;
          then o,c // o,d2 by A327,A328,AFF_1:5;
          then
A333:     LIN o,c,d2 by AFF_1:def 1;
          take d2;
          d,e2 // d,d2 by A332,AFF_1:def 1;
          hence thesis by A5,A8,A13,A17,A329,A330,A333,AFF_1:5,25;
        end;
        then consider d2 such that
A334:   a,c // d,d2 and
A335:   d2 in M;
        consider d1 such that
A336:   a,b // d,d1 and
A337:   d1 in Z by A288;
        assume
A338:   not Z // M;
A339:   d1<>d2
        proof
A340:     o<>d1
          proof
A341:       o<>d
            proof
              assume o=d;
              then LIN o,b,c by A280,AFF_1:6;
              then c in Z by A4,A7,A11,A16,AFF_1:25;
              then
A342:         o,c // Z by A4,A16,AFF_1:52;
              o,c // M by A5,A13,A17,AFF_1:52;
              hence contradiction by A8,A338,A342,AFF_1:53;
            end;
A343:       a,a1 // a,d by A282,AFF_1:def 1;
            a,o // a,a1 by A3,A9,A10,A15,AFF_1:39,41;
            then a,o // a,d by A297,A343,AFF_1:5;
            then LIN a,o,d by AFF_1:def 1;
            then LIN o,a,d by AFF_1:6;
            then o,a // o,d by AFF_1:def 1;
            then
A344:       a,o // d,o by AFF_1:4;
            assume o=d1;
            then a,b // a,o by A336,A341,A344,AFF_1:5;
            then LIN a,b,o by AFF_1:def 1;
            then LIN o,b,a by AFF_1:6;
            hence contradiction by A4,A7,A11,A16,A283,AFF_1:25;
          end;
          assume d1=d2;
          then
A345:     o,d1 // M by A5,A17,A335,AFF_1:52;
          o,d1 // Z by A4,A16,A337,AFF_1:52;
          hence contradiction by A338,A345,A340,AFF_1:53;
        end;
A346:   now
          assume
A347:     b1,c1 // d1,d2;
          ex d5 st LIN b,c,d5 & LIN d1,d2,d5
          proof
            consider e1 such that
A348:       b,c // b,e1 and
A349:       b<>e1 by DIRAF:40;
            consider e2 such that
A350:       d1,d2 // d1,e2 and
A351:       d1<>e2 by DIRAF:40;
            not b,e1 // d1,e2
            proof
              assume b,e1 // d1,e2;
              then b,e1 // d1,d2 by A350,A351,AFF_1:5;
              then b,c // d1,d2 by A348,A349,AFF_1:5;
              hence contradiction by A22,A339,A347,AFF_1:5;
            end;
            then consider d5 such that
A352:       LIN b,e1,d5 and
A353:       LIN d1,e2,d5 by AFF_1:60;
            b,e1 // b,d5 by A352,AFF_1:def 1;
            then
A354:       b,c // b,d5 by A348,A349,AFF_1:5;
            take d5;
            d1,e2 // d1,d5 by A353,AFF_1:def 1;
            then d1,d2 // d1,d5 by A350,A351,AFF_1:5;
            hence thesis by A354,AFF_1:def 1;
          end;
          then consider d5 such that
A355:     LIN b,c,d5 and
A356:     LIN d1,d2,d5;
A357:     d in Y by A9,A10,A15,A297,A282,AFF_1:25;
A358:     now
A359:       not LIN a,b,d
            proof
A360:         a<>d
              proof
A361:           a1<>b1
                proof
                  o,a1 // o,a by A3,A9,A10,A15,AFF_1:39,41;
                  then
A362:             LIN o,a1,a by AFF_1:def 1;
                  assume a1=b1;
                  hence contradiction by A4,A12,A16,A283,A307,A362,AFF_1:25;
                end;
                assume
A363:           a=d;
                then LIN a,b,c by A280,AFF_1:6;
                then a,b // a,c by AFF_1:def 1;
                then a1,b1 // a,c by A11,A20,A283,AFF_1:5;
                then a1,b1 // a1,c1 by A13,A21,A283,AFF_1:5;
                then LIN a1,b1,c1 by AFF_1:def 1;
                then LIN b1,c1,a1 by AFF_1:6;
                then b1,c1 // b1,a1 by AFF_1:def 1;
                then
A364:           b1,c1 // a1,b1 by AFF_1:4;
                b,c // b,a by A280,A363,AFF_1:def 1;
                then b,c // a,b by AFF_1:4;
                then b,c // a1,b1 by A11,A20,A283,AFF_1:5;
                hence contradiction by A22,A364,A361,AFF_1:5;
              end;
              assume LIN a,b,d;
              then LIN a,d,b by AFF_1:6;
              then b in Y by A9,A15,A357,A360,AFF_1:25;
              then
A365:         o,b // Y by A3,A15,AFF_1:52;
              o,b // Z by A4,A11,A16,AFF_1:52;
              hence contradiction by A3,A4,A7,A18,A365,AFF_1:45,53;
            end;
            assume
A366:       LIN a,d,d5;
A367:       o<>d
            proof
A368:         o,b // o,b1 by A4,A11,A12,A16,AFF_1:39,41;
              assume
A369:         o=d;
              then LIN o,b,c by A280,AFF_1:6;
              then o,b // o,c by AFF_1:def 1;
              then
A370:         o,b1 // o,c by A7,A368,AFF_1:5;
              o,c // o,c1 by A5,A13,A14,A17,AFF_1:39,41;
              then o,b1 // o,c1 by A8,A370,AFF_1:5;
              then LIN o,b1,c1 by AFF_1:def 1;
              then LIN b1,c1,o by AFF_1:6;
              then b1,c1 // b1,o by AFF_1:def 1;
              then
A371:         b1,c1 // o,b1 by AFF_1:4;
              b,c // b,o by A280,A369,AFF_1:def 1;
              then b,c // o,b by AFF_1:4;
              then b,c // o,b1 by A7,A368,AFF_1:5;
              hence contradiction by A22,A307,A371,AFF_1:5;
            end;
A372:       d<>d1
            proof
              assume d=d1;
              then
A373:         o,d // Z by A4,A16,A337,AFF_1:52;
              o,d // Y by A3,A15,A357,AFF_1:52;
              hence contradiction by A3,A4,A18,A367,A373,AFF_1:45,53;
            end;
A374:       d<>d2
            proof
              assume d=d2;
              then
A375:         o,d // M by A5,A17,A335,AFF_1:52;
              o,d // Y by A3,A15,A357,AFF_1:52;
              hence contradiction by A3,A5,A19,A367,A375,AFF_1:45,53;
            end;
A376:       b,c // b,d5 by A355,AFF_1:def 1;
A377:       b,c // b,d by A280,AFF_1:def 1;
            b<>c by A22,AFF_1:3;
            then b,d // b,d5 by A377,A376,AFF_1:5;
            then LIN d1,d2,d by A356,A366,A359,AFF_1:14;
            then LIN d,d1,d2 by AFF_1:6;
            then d,d1 // d,d2 by AFF_1:def 1;
            then a,b // d,d2 by A336,A372,AFF_1:5;
            then
A378:       a,b // a,c by A334,A374,AFF_1:5;
            then LIN a,b,c by AFF_1:def 1;
            then LIN b,c,a by AFF_1:6;
            then b,c // b, a by AFF_1:def 1;
            then b,c // a,b by AFF_1:4;
            then
A379:       b,c // a1,b1 by A11,A20,A283,AFF_1:5;
            a1,b1 // a,c by A11,A20,A283,A378,AFF_1:5;
            then a1,b1 // a1,c1 by A13,A21,A283,AFF_1:5;
            then LIN a1,b1,c1 by AFF_1:def 1;
            then LIN b1,c1,a1 by AFF_1:6;
            then b1,c1 // b1,a1 by AFF_1:def 1;
            then
A380:       b1,c1 // a1,b1 by AFF_1:4;
            a1<>b1
            proof
              o,a1 // o,a by A3,A9,A10,A15,AFF_1:39,41;
              then
A381:         LIN o,a1,a by AFF_1:def 1;
              assume a1=b1;
              hence contradiction by A4,A12,A16,A283,A307,A381,AFF_1:25;
            end;
            hence contradiction by A22,A380,A379,AFF_1:5;
          end;
          not b,c // d1,d2 by A22,A339,A347,AFF_1:5;
          hence
          contradiction by A2,A3,A4,A5,A6,A7,A8,A9,A11,A13,A15,A16,A17,A18,A19
,A23,A336,A337,A334,A335,A355,A356,A357,A358;
        end;
        now
A382:     d in Y by A9,A10,A15,A297,A282,AFF_1:25;
A383:     not LIN a1,b1,d
          proof
A384:       a1<>d
            proof
A385:         a1<>b1 & a1<>c1
              proof
                o,a1 // o,a by A3,A9,A10,A15,AFF_1:39,41;
                then
A386:           LIN o,a1,a by AFF_1:def 1;
                assume not thesis;
                hence contradiction by A4,A5,A12,A14,A16,A17,A283,A307,A386,
AFF_1:25;
              end;
              assume
A387:         a1=d;
              then LIN a1,b1,c1 by A281,AFF_1:6;
              then a1,b1 // a1,c1 by AFF_1:def 1;
              then a,b // a1,c1 by A20,A385,AFF_1:5;
              then a,b // a,c by A21,A385,AFF_1:5;
              then LIN a,b,c by AFF_1:def 1;
              then LIN b,c,a by AFF_1:6;
              then b,c // b,a by AFF_1:def 1;
              then
A388:         b,c // a,b by AFF_1:4;
              b1,c1 // b1,a1 by A281,A387,AFF_1:def 1;
              then b1,c1 // a1,b1 by AFF_1:4;
              then b1,c1 // a,b by A20,A385,AFF_1:5;
              hence contradiction by A11,A22,A283,A388,AFF_1:5;
            end;
            assume LIN a1,b1,d;
            then LIN a1,d,b1 by AFF_1:6;
            then b1 in Y by A10,A15,A382,A384,AFF_1:25;
            then o,b1 // o,a by A3,A9,A15,AFF_1:39,41;
            then LIN o,b1,a by AFF_1:def 1;
            hence contradiction by A4,A12,A16,A283,A307,AFF_1:25;
          end;
A389:     d<>o
          proof
A390:       o,b // o,b1 by A4,A11,A12,A16,AFF_1:39,41;
            assume
A391:       d=o;
            then LIN o,b,c by A280,AFF_1:6;
            then o,b // o,c by AFF_1:def 1;
            then
A392:       o,b1 // o,c by A7,A390,AFF_1:5;
            o,c // o,c1 by A5,A13,A14,A17,AFF_1:39,41;
            then o,b1 // o,c1 by A8,A392,AFF_1:5;
            then LIN o,b1,c1 by AFF_1:def 1;
            then LIN b1,c1,o by AFF_1:6;
            then b1,c1 // b1,o by AFF_1:def 1;
            then
A393:       b1,c1 // o,b1 by AFF_1:4;
            b,c // b,o by A280,A391,AFF_1:def 1;
            then b,c // o,b by AFF_1:4;
            then b,c // o,b1 by A7,A390,AFF_1:5;
            hence contradiction by A22,A307,A393,AFF_1:5;
          end;
A394:     d<>d1
          proof
            o,d // o,a by A3,A9,A15,A382,AFF_1:39,41;
            then
A395:       LIN o,d,a by AFF_1:def 1;
            assume d=d1;
            hence contradiction by A4,A16,A283,A337,A389,A395,AFF_1:25;
          end;
A396:     d<>d2
          proof
            o,d // o,a by A3,A9,A15,A382,AFF_1:39,41;
            then
A397:       LIN o,d,a by AFF_1:def 1;
            assume d=d2;
            hence contradiction by A5,A17,A283,A335,A389,A397,AFF_1:25;
          end;
A398:     a1,c1 // d,d2 by A13,A21,A283,A334,AFF_1:5;
A399:     b1<>c1 by A22,AFF_1:3;
A400:     b1,c1 // b1,d by A281,AFF_1:def 1;
          assume
A401:     not b1,c1 // d1,d2;
          then consider d5 such that
A402:     LIN b1,c1,d5 and
A403:     LIN d1,d2,d5 by AFF_1:60;
          b1,c1 // b1,d5 by A402,AFF_1:def 1;
          then
A404:     b1,d // b1,d5 by A399,A400,AFF_1:5;
          a1,b1 // d,d1 by A11,A20,A283,A336,AFF_1:5;
          then LIN a1,d,d5 by A2,A3,A4,A5,A10,A12,A14,A15,A16,A17,A18,A19,A23
,A307,A337,A335,A401,A402,A403,A382,A398;
          then d=d5 by A383,A404,AFF_1:14;
          then LIN d,d1,d2 by A403,AFF_1:6;
          then d,d1 // d,d2 by AFF_1:def 1;
          then a,b // d,d2 by A336,A394,AFF_1:5;
          then
A405:     a,b // a,c by A334,A396,AFF_1:5;
          then LIN a,b,c by AFF_1:def 1;
          then LIN b,c,a by AFF_1:6;
          then b,c // b, a by AFF_1:def 1;
          then b,c // a,b by AFF_1:4;
          then
A406:     b,c // a1,b1 by A11,A20,A283,AFF_1:5;
          a1,b1 // a,c by A11,A20,A283,A405,AFF_1:5;
          then a1,b1 // a1,c1 by A13,A21,A283,AFF_1:5;
          then LIN a1,b1,c1 by AFF_1:def 1;
          then LIN b1,c1,a1 by AFF_1:6;
          then b1,c1 // b1,a1 by AFF_1:def 1;
          then
A407:     b1,c1 // a1,b1 by AFF_1:4;
          a1<>b1
          proof
            o,a1 // o,a by A3,A9,A10,A15,AFF_1:39,41;
            then
A408:       LIN o,a1,a by AFF_1:def 1;
            assume a1=b1;
            hence contradiction by A4,A12,A16,A283,A307,A408,AFF_1:25;
          end;
          hence contradiction by A22,A407,A406,AFF_1:5;
        end;
        hence contradiction by A346;
      end;
      now
        assume
A409:   Z // M;
        then
A410:   b1 in M by A4,A5,A12,AFF_1:45;
        b in M by A4,A5,A11,A409,AFF_1:45;
        hence contradiction by A13,A14,A17,A22,A410,AFF_1:39,41;
      end;
      hence contradiction by A278;
    end;
    hence thesis by AFF_2:def 4;
  end;
  X is Desarguesian implies X is satisfying_Scherungssatz
  proof
    assume
A411: X is Desarguesian;
    then X is satisfying_TDES_1 by AFF_2:3,12;
    then X is satisfying_des_1 by AFF_2:13;
    then X is translational by AFF_2:7;
    then
A412: X is satisfying_minor_Scherungssatz by Th6;
    X is satisfying_major_Scherungssatz by A411,Th7;
    hence thesis by A412,Th2;
  end;
  hence thesis by A1;
end;
