reserve X for OrtAfPl;
reserve o,a,a1,a2,a3,a4,b,b1,b2,b3,b4,c,c1,c2,c3,d,d1,d2,d3,d4,e1,e2 for
  Element of X;
reserve a29,a39,b29,x9 for Element of the AffinStruct of X;
reserve A,K,M,N for Subset of X;
reserve A9,K9 for Subset of the AffinStruct of X;

theorem
  X is satisfying_OPAP & X is satisfying_DES implies X is satisfying_PAP
proof
  assume that
A1: X is satisfying_OPAP and
A2: X is satisfying_DES;
  let o,a1,a2,a3,b1,b2,b3,M,N;
  assume that
A3: M is being_line and
A4: N is being_line and
A5: o in M and
A6: a1 in M and
A7: a2 in M and
A8: a3 in M and
A9: o in N and
A10: b1 in N and
A11: b2 in N and
A12: b3 in N and
A13: not b2 in M and
A14: not a3 in N and
A15: o<>a1 and
A16: o<>a2 and
  o<>a3 and
A17: o<>b1 and
  o<>b2 and
A18: o<>b3 and
A19: a3,b2 // a2,b1 and
A20: a3,b3 // a1,b1;
  reconsider o9=o,a19=a1,a29=a2,a39=a3,b19=b1,b29=b2,b39=b3
  as Element of the AffinStruct of X;
  reconsider M9=M,N9=N as Subset of the AffinStruct of X;
A21: M9 is being_line by A3,ANALMETR:43;
A22: N9 is being_line by A4,ANALMETR:43;
  now
    assume
A23: not M _|_ N;
A24: now
      assume that
A25:  a1<>a2 and
A26:  a2<>a3 and
A27:  a1<>a3;
A28:  b1<>b2 & b2<>b3 & b1<>b3
      proof
A29:    now
A30:      not LIN o9,b19,a19
          proof
            o9,a19 // o9,a39 by A5,A6,A8,A21,AFF_1:39,41;
            then
A31:        LIN o9,a19,a39 by AFF_1:def 1;
            assume LIN o9,b19,a19;
            then a19 in N9 by A9,A10,A17,A22,AFF_1:25;
            hence contradiction by A9,A14,A15,A22,A31,AFF_1:25;
          end;
          assume b1=b3;
          then b1,a1 // b1,a3 by A20,ANALMETR:59;
          then
A32:      b19,a19 // b19,a39 by ANALMETR:36;
          o9,a19 // o9,a39 by A5,A6,A8,A21,AFF_1:39,41;
          then LIN o9,a19,a39 by AFF_1:def 1;
          hence contradiction by A27,A30,A32,AFF_1:14;
        end;
A33:    now
A34:      not LIN o9,b19,a19
          proof
            o9,a19 // o9,a39 by A5,A6,A8,A21,AFF_1:39,41;
            then
A35:        LIN o9,a19,a39 by AFF_1:def 1;
            assume LIN o9,b19,a19;
            then a19 in N9 by A9,A10,A17,A22,AFF_1:25;
            hence contradiction by A9,A14,A15,A22,A35,AFF_1:25;
          end;
          assume b2=b3;
          then a1,b1 // a2,b1 by A8,A13,A19,A20,ANALMETR:60;
          then b1,a1 // b1,a2 by ANALMETR:59;
          then
A36:      b19,a19 // b19,a29 by ANALMETR:36;
          o9,a19 // o9,a29 by A5,A6,A7,A21,AFF_1:39,41;
          then LIN o9,a19,a29 by AFF_1:def 1;
          hence contradiction by A25,A34,A36,AFF_1:14;
        end;
A37:    now
A38:      not LIN o9,b29,a29
          proof
            assume LIN o9,b29,a29;
            then LIN o9,a29,b29 by AFF_1:6;
            hence contradiction by A5,A7,A13,A16,A21,AFF_1:25;
          end;
          assume b1=b2;
          then b2,a2 // b2,a3 by A19,ANALMETR:59;
          then
A39:      b29,a29 // b29,a39 by ANALMETR:36;
          o9,a29 // o9,a39 by A5,A7,A8,A21,AFF_1:39,41;
          then LIN o9,a29,a39 by AFF_1:def 1;
          hence contradiction by A26,A38,A39,AFF_1:14;
        end;
        assume b1=b2 or b2=b3 or b1=b3;
        hence contradiction by A37,A33,A29;
      end;
A40:  now
A41:    not LIN b3,b1,a3
        proof
          assume LIN b3,b1,a3;
          then LIN b39,b19,a39 by ANALMETR:40;
          hence contradiction by A10,A12,A14,A22,A28,AFF_1:25;
        end;
        o9,b29 // o9,b39 by A9,A11,A12,A22,AFF_1:39,41;
        then LIN o9,b29,b39 by AFF_1:def 1;
        then
A42:    LIN o,b2,b3 by ANALMETR:40;
A43:    not LIN b2,b1,a3
        proof
          assume LIN b2,b1,a3;
          then LIN b29,b19,a39 by ANALMETR:40;
          hence contradiction by A10,A11,A14,A22,A28,AFF_1:25;
        end;
        o9,a39 // o9,a19 by A5,A6,A8,A21,AFF_1:39,41;
        then LIN o9,a39,a19 by AFF_1:def 1;
        then
A44:    LIN o,a3,a1 by ANALMETR:40;
        consider x be Element of X such that
A45:    o,b2 _|_ o,x and
A46:    o<>x by ANALMETR:39;
A47:    o,x _|_ N by A4,A5,A9,A11,A13,A45,ANALMETR:55;
        consider e19 be Element of the AffinStruct of X such that
A48:    a39,b39 // b29,e19 and
A49:    b29<> e19 by DIRAF:40;
        reconsider x9=x as Element of the AffinStruct of X;
        LIN o9,x9,x9 by AFF_1:7;
        then consider A9 be Subset of the AffinStruct of X such that
A50:    A9 is being_line and
A51:    o9 in A9 and
A52:    x9 in A9 and
        x9 in A9 by AFF_1:21;
        reconsider A=A9 as Subset of X;
A53:    for e1 holds e1 in A implies LIN o,x,e1
        proof
          let e1 such that
A54:      e1 in A;
          reconsider e19=e1 as Element of the AffinStruct of X;
          o9,x9 // o9,e19 by A50,A51,A52,A54,AFF_1:39,41;
          then LIN o9,x9,e19 by AFF_1:def 1;
          hence thesis by ANALMETR:40;
        end;
        for e1 holds LIN o,x,e1 implies e1 in A
        proof
          let e1 such that
A55:      LIN o,x,e1;
          reconsider e19=e1 as Element of the AffinStruct of X;
          LIN o9,x9,e19 by A55,ANALMETR:40;
          hence thesis by A46,A50,A51,A52,AFF_1:25;
        end;
        then A=Line(o,x) by A53,ANALMETR:def 11;
        then
A56:    A _|_ N by A46,A47,ANALMETR:def 14;
A57:    not b2 in A
        proof
A58:      o9,b29 // o9,b29 by AFF_1:2;
          assume b2 in A;
          then A9 // N9 by A5,A9,A11,A13,A22,A50,A51,A58,AFF_1:38;
          then A // N by ANALMETR:46;
          hence contradiction by A56,ANALMETR:52;
        end;
        assume
A59:    not a3,b3 _|_ o,b2;
        not b29,e19 // A9
        proof
          assume
A60:      b29,e19 // A9;
          consider d19,d29 be Element of the AffinStruct of X such that
A61:      d19<>d29 and
A62:      A9=Line(d19,d29) by A50,AFF_1:def 3;
A63:      d29 in A9 by A62,AFF_1:15;
          reconsider d1=d19,d2=d29 as Element of X;
          d19,d29 // d19,d29 by AFF_1:2;
          then d19,d29 // A9 by A61,A62,AFF_1:def 4;
          then b29,e19 // d19,d29 by A50,A60,AFF_1:31;
          then a39,b39 // d19,d29 by A48,A49,AFF_1:5;
          then
A64:      a3,b3 // d1,d2 by ANALMETR:36;
          d19 in A9 by A62,AFF_1:15;
          then d1,d2 _|_ o,b2 by A9,A11,A56,A63,ANALMETR:56;
          hence contradiction by A59,A61,A64,ANALMETR:62;
        end;
        then consider c39 be Element of the AffinStruct of X such that
A65:    c39 in A9 and
A66:    LIN b29,e19,c39 by A50,AFF_1:59;
        b29,e19 // b29,c39 by A66,AFF_1:def 1;
        then
A67:    a39,b39 // b29,c39 by A48,A49,AFF_1:5;
        consider e19 be Element of the AffinStruct of X such that
A68:    c39,a39 // a19,e19 and
A69:    a19<> e19 by DIRAF:40;
        not a19,e19 // A9
        proof
A70:      c39<>o9
          proof
            assume c39=o9;
            then
A71:        a3,b3 // b2,o by A67,ANALMETR:36;
            b29,o9 // b29,b39 by A9,A11,A12,A22,AFF_1:39,41;
            then b2,o // b2,b3 by ANALMETR:36;
            then a3,b3 // b2,b3 by A5,A13,A71,ANALMETR:60;
            then b3,b2 // b3,a3 by ANALMETR:59;
            then LIN b3,b2,a3 by ANALMETR:def 10;
            then LIN b39,b29,a39 by ANALMETR:40;
            hence contradiction by A11,A12,A14,A22,A28,AFF_1:25;
          end;
          assume
A72:      a19,e19 // A9;
A73:      o9,a39 // o9,a39 by AFF_1:2;
          o9,c39 // A9 by A50,A51,A65,AFF_1:40,41;
          then a19,e19 // o9,c39 by A50,A72,AFF_1:31;
          then c39,a39 // o9,c39 by A68,A69,AFF_1:5;
          then c39,o9 // c39,a39 by AFF_1:4;
          then LIN c39,o9,a39 by AFF_1:def 1;
          then a39 in A9 by A50,A51,A65,A70,AFF_1:25;
          then A9 // M9 by A5,A8,A9,A14,A21,A50,A51,A73,AFF_1:38;
          then A // M by ANALMETR:46;
          hence contradiction by A23,A56,ANALMETR:52;
        end;
        then consider c19 be Element of the AffinStruct of X such that
A74:    c19 in A9 and
A75:    LIN a19,e19,c19 by A50,AFF_1:59;
        reconsider c1=c19 as Element of X;
        reconsider c3=c39 as Element of X;
        a19,e19 // a19,c19 by A75,AFF_1:def 1;
        then
A76:    c39,a39 // a19,c19 by A68,A69,AFF_1:5;
        then
A77:    c3,a3 // a1,c1 by ANALMETR:36;
        consider e19 be Element of the AffinStruct of X such that
A78:    c39,a39 // a29,e19 and
A79:    a29<> e19 by DIRAF:40;
        not a29,e19 // A9
        proof
A80:      c39<>o9
          proof
            assume c39=o9;
            then
A81:        a3,b3 // b2,o by A67,ANALMETR:36;
            b29,o9 // b29,b39 by A9,A11,A12,A22,AFF_1:39,41;
            then b2,o // b2,b3 by ANALMETR:36;
            then a3,b3 // b2,b3 by A5,A13,A81,ANALMETR:60;
            then b3,b2 // b3,a3 by ANALMETR:59;
            then LIN b3,b2,a3 by ANALMETR:def 10;
            then LIN b39,b29,a39 by ANALMETR:40;
            hence contradiction by A11,A12,A14,A22,A28,AFF_1:25;
          end;
          assume
A82:      a29,e19 // A9;
A83:      o9,a39 // o9,a39 by AFF_1:2;
          o9,c39 // A9 by A50,A51,A65,AFF_1:40,41;
          then a29,e19 // o9,c39 by A50,A82,AFF_1:31;
          then c39,a39 // o9,c39 by A78,A79,AFF_1:5;
          then c39,o9 // c39,a39 by AFF_1:4;
          then LIN c39,o9,a39 by AFF_1:def 1;
          then a39 in A9 by A50,A51,A65,A80,AFF_1:25;
          then A9 // M9 by A5,A8,A9,A14,A21,A50,A51,A83,AFF_1:38;
          then A // M by ANALMETR:46;
          hence contradiction by A23,A56,ANALMETR:52;
        end;
        then consider c29 be Element of the AffinStruct of X such that
A84:    c29 in A9 and
A85:    LIN a29,e19,c29 by A50,AFF_1:59;
        reconsider c2=c29 as Element of X;
        a29,e19 // a29,c29 by A85,AFF_1:def 1;
        then
A86:    c39,a39 // a29,c29 by A78,A79,AFF_1:5;
        then
A87:    c3,a3 // a2,c2 by ANALMETR:36;
A88:    o<>c3 & o<>c2 & o<>c1
        proof
A89:      now
            assume
A90:        o=c3;
            b29,o9 // b39,b29 by A9,A11,A12,A22,AFF_1:39,41;
            then a39,b39 // b39,b29 by A5,A13,A67,A90,AFF_1:5;
            then b39,b29 // b39,a39 by AFF_1:4;
            then LIN b39,b29,a39 by AFF_1:def 1;
            hence contradiction by A11,A12,A14,A22,A28,AFF_1:25;
          end;
A91:      now
            a19,o9 // a19,a39 by A5,A6,A8,A21,AFF_1:39,41;
            then
A92:        a1,o // a1,a3 by ANALMETR:36;
            assume o=c2 or o=c1;
            then
A93:        c3,a3 // a1,o or c3,a3 // a2,o by A76,A86,ANALMETR:36;
            a29,o9 // a19,a39 by A5,A6,A7,A8,A21,AFF_1:39,41;
            then a2,o // a1,a3 by ANALMETR:36;
            then c3,a3 // a1,a3 by A15,A16,A93,A92,ANALMETR:60;
            then a3,a1 // a3,c3 by ANALMETR:59;
            then LIN a3,a1,c3 by ANALMETR:def 10;
            then LIN a39,a19,c39 by ANALMETR:40;
            then
A94:        c39 in M9 by A6,A8,A21,A27,AFF_1:25;
            o9,c39 // o9,c39 by AFF_1:2;
            then A9 // M9 by A5,A21,A50,A51,A65,A89,A94,AFF_1:38;
            then A // M by ANALMETR:46;
            hence contradiction by A23,A56,ANALMETR:52;
          end;
          assume o=c3 or o=c2 or o=c1;
          hence contradiction by A89,A91;
        end;
A95:    not c3 in N
        proof
A96:      o9,c39 // o9,c39 by AFF_1:2;
          assume c3 in N;
          then A9 // N9 by A9,A22,A50,A51,A65,A88,A96,AFF_1:38;
          then A // N by ANALMETR:46;
          hence contradiction by A56,ANALMETR:52;
        end;
A97:    not LIN a3,a2,c3
        proof
          assume LIN a3,a2,c3;
          then LIN a39,a29,c39 by ANALMETR:40;
          then
A98:      c39 in M9 by A7,A8,A21,A26,AFF_1:25;
          o9,c39 // o9,c39 by AFF_1:2;
          then A9 // M9 by A5,A21,A50,A51,A65,A88,A98,AFF_1:38;
          then A // M by ANALMETR:46;
          hence contradiction by A23,A56,ANALMETR:52;
        end;
A99:    not LIN a3,a1,c3
        proof
          assume LIN a3,a1,c3;
          then LIN a39,a19,c39 by ANALMETR:40;
          then
A100:     c39 in M9 by A6,A8,A21,A27,AFF_1:25;
          o9,c39 // o9,c39 by AFF_1:2;
          then A9 // M9 by A5,A21,A50,A51,A65,A88,A100,AFF_1:38;
          then A // M by ANALMETR:46;
          hence contradiction by A23,A56,ANALMETR:52;
        end;
A101:   not LIN a1,a2,c1
        proof
          assume LIN a1,a2,c1;
          then LIN a19,a29,c19 by ANALMETR:40;
          then
A102:     c19 in M9 by A6,A7,A21,A25,AFF_1:25;
          o9,c19 // o9,c19 by AFF_1:2;
          then M9 // A9 by A5,A21,A50,A51,A74,A88,A102,AFF_1:38;
          then M // A by ANALMETR:46;
          hence contradiction by A23,A56,ANALMETR:52;
        end;
        o9,a19 // o9,a29 by A5,A6,A7,A21,AFF_1:39,41;
        then LIN o9,a19,a29 by AFF_1:def 1;
        then
A103:   LIN o,a1,a2 by ANALMETR:40;
        o9,b29 // o9,b19 by A9,A10,A11,A22,AFF_1:39,41;
        then LIN o9,b29,b19 by AFF_1:def 1;
        then
A104:   LIN o,b2,b1 by ANALMETR:40;
        o9,c19 // o9,c29 by A50,A51,A74,A84,AFF_1:39,41;
        then LIN o9,c19,c29 by AFF_1:def 1;
        then
A105:   LIN o,c1,c2 by ANALMETR:40;
        o9,c39 // o9,c29 by A50,A51,A65,A84,AFF_1:39,41;
        then LIN o9,c39,c29 by AFF_1:def 1;
        then
A106:   LIN o,c3,c2 by ANALMETR:40;
        o9,c39 // o9,c19 by A50,A51,A65,A74,AFF_1:39,41;
        then LIN o9,c39,c19 by AFF_1:def 1;
        then
A107:   LIN o,c3,c1 by ANALMETR:40;
        c3<>a3
        proof
A108:     o9,a39 // o9,a39 by AFF_1:2;
          assume c3=a3;
          then A9 // M9 by A5,A8,A9,A14,A21,A50,A51,A65,A108,AFF_1:38;
          then A // M by ANALMETR:46;
          hence contradiction by A23,A56,ANALMETR:52;
        end;
        then a1,c1 // a2,c2 by A77,A87,ANALMETR:60;
        then
A109:   c1,a1 // c2,a2 by ANALMETR:59;
A110:   not LIN c1,c2,b2
        proof
A111:     c19<>c29
          proof
A112:       c3<>a3
            proof
A113:         o9,c39 // o9,c39 by AFF_1:2;
              assume c3=a3;
              then A9 // M9 by A5,A8,A9,A14,A21,A50,A51,A65,A113,AFF_1:38;
              then A // M by ANALMETR:46;
              hence contradiction by A23,A56,ANALMETR:52;
            end;
            assume c19=c29;
            then a1,c1 // a2,c1 by A77,A87,A112,ANALMETR:60;
            then c1,a2 // c1,a1 by ANALMETR:59;
            then LIN c1,a2,a1 by ANALMETR:def 10;
            then LIN c19,a29,a19 by ANALMETR:40;
            then LIN a19,a29,c19 by AFF_1:6;
            then
A114:       c19 in M9 by A6,A7,A21,A25,AFF_1:25;
            o9,c19 // o9,c19 by AFF_1:2;
            then A9 // M9 by A5,A21,A50,A51,A74,A88,A114,AFF_1:38;
            then A // M by ANALMETR:46;
            hence contradiction by A23,A56,ANALMETR:52;
          end;
          assume LIN c1,c2,b2;
          then LIN c19,c29,b29 by ANALMETR:40;
          then b29 in A9 by A50,A74,A84,A111,AFF_1:25;
          then A9=N9 by A5,A9,A11,A13,A22,A50,A51,AFF_1:18;
          then A9 // N9 by A22,AFF_1:41;
          then A // N by ANALMETR:46;
          hence contradiction by A56,ANALMETR:52;
        end;
        o9,a39 // o9,a29 by A5,A7,A8,A21,AFF_1:39,41;
        then LIN o9,a39,a29 by AFF_1:def 1;
        then
A115:   LIN o,a3,a2 by ANALMETR:40;
        o9,b39 // o9,b19 by A9,A10,A12,A22,AFF_1:39,41;
        then LIN o9,b39,b19 by AFF_1:def 1;
        then
A116:   LIN o,b3,b1 by ANALMETR:40;
        a3,c3 // a1,c1 by A77,ANALMETR:59;
        then b3,c3 // b1,c1 by A2,A9,A14,A15,A17,A18,A20,A88,A99,A41,A44,A116
,A107,CONAFFM:def 1;
        then
A117:   c3,b3 // c1,b1 by ANALMETR:59;
        a3,c3 // a2,c2 by A87,ANALMETR:59;
        then b2,c3 // b1,c2 by A2,A5,A9,A13,A14,A16,A17,A19,A88,A115,A104,A106
,A43,A97,CONAFFM:def 1;
        then c3,b2 // c2,b1 by ANALMETR:59;
        then c1,b2 // c2,b3 by A1,A9,A10,A11,A12,A17,A18,A51,A56,A65,A74,A84
,A88,A117,A57,A95;
        hence thesis by A2,A5,A13,A15,A16,A18,A88,A103,A42,A105,A101,A110,A109,
CONAFFM:def 1;
      end;
      now
        o9,a19 // o9,a29 by A5,A6,A7,A21,AFF_1:39,41;
        then LIN o9,a19,a29 by AFF_1:def 1;
        then
A118:   LIN o,a1,a2 by ANALMETR:40;
        o9,b29 // o9,b19 by A9,A10,A11,A22,AFF_1:39,41;
        then LIN o9,b29,b19 by AFF_1:def 1;
        then
A119:   LIN o,b2,b1 by ANALMETR:40;
        o9,b29 // o9,b39 by A9,A11,A12,A22,AFF_1:39,41;
        then LIN o9,b29,b39 by AFF_1:def 1;
        then
A120:   LIN o,b2,b3 by ANALMETR:40;
A121:   not LIN b3,b1,a3 & not LIN b2,b1,a3
        proof
          assume LIN b3,b1,a3 or LIN b2,b1,a3;
          then LIN b39,b19,a39 or LIN b29,b19,a39 by ANALMETR:40;
          hence contradiction by A10,A11,A12,A14,A22,A28,AFF_1:25;
        end;
        o9,a39 // o9,a29 by A5,A7,A8,A21,AFF_1:39,41;
        then LIN o9,a39,a29 by AFF_1:def 1;
        then
A122:   LIN o,a3,a2 by ANALMETR:40;
        o9,b39 // o9,b19 by A9,A10,A12,A22,AFF_1:39,41;
        then LIN o9,b39,b19 by AFF_1:def 1;
        then
A123:   LIN o,b3,b1 by ANALMETR:40;
        o9,a39 // o9,a19 by A5,A6,A8,A21,AFF_1:39,41;
        then LIN o9,a39,a19 by AFF_1:def 1;
        then
A124:   LIN o,a3,a1 by ANALMETR:40;
        consider x be Element of X such that
A125:   o,b2 _|_ o,x and
A126:   o<>x by ANALMETR:39;
A127:   o,x _|_ N by A4,A5,A9,A11,A13,A125,ANALMETR:55;
        consider e19 be Element of the AffinStruct of X such that
A128:   a39,b29 // b39,e19 and
A129:   b39<> e19 by DIRAF:40;
        reconsider x9=x as Element of the AffinStruct of X;
        LIN o9,x9,x9 by AFF_1:7;
        then consider A9 be Subset of the AffinStruct of X such that
A130:   A9 is being_line and
A131:   o9 in A9 and
A132:   x9 in A9 and
        x9 in A9 by AFF_1:21;
        reconsider A=A9 as Subset of X;
A133:   for e1 holds e1 in A implies LIN o,x,e1
        proof
          let e1 such that
A134:     e1 in A;
          reconsider e19=e1 as Element of the AffinStruct of X;
          o9,x9 // o9,e19 by A130,A131,A132,A134,AFF_1:39,41;
          then LIN o9,x9,e19 by AFF_1:def 1;
          hence thesis by ANALMETR:40;
        end;
        for e1 holds LIN o,x,e1 implies e1 in A
        proof
          let e1 such that
A135:     LIN o,x,e1;
          reconsider e19=e1 as Element of the AffinStruct of X;
          LIN o9,x9,e19 by A135,ANALMETR:40;
          hence thesis by A126,A130,A131,A132,AFF_1:25;
        end;
        then A=Line(o,x) by A133,ANALMETR:def 11;
        then
A136:   A _|_ N by A126,A127,ANALMETR:def 14;
        assume
A137:   a3,b3 _|_ o,b2;
        not b39,e19 // A9
        proof
          assume
A138:     b39,e19 // A9;
          consider d19,d29 be Element of the AffinStruct of X such that
A139:     d19<>d29 and
A140:     A9=Line(d19,d29) by A130,AFF_1:def 3;
A141:     d29 in A9 by A140,AFF_1:15;
          reconsider d1=d19,d2=d29 as Element of X;
          d19,d29 // d19,d29 by AFF_1:2;
          then d19,d29 // A9 by A139,A140,AFF_1:def 4;
          then b39,e19 // d19,d29 by A130,A138,AFF_1:31;
          then a39,b29 // d19,d29 by A128,A129,AFF_1:5;
          then
A142:     a3,b2 // d1,d2 by ANALMETR:36;
          d19 in A9 by A140,AFF_1:15;
          then d1,d2 _|_ o,b2 by A9,A11,A136,A141,ANALMETR:56;
          then a3,b2 _|_ o,b2 by A139,A142,ANALMETR:62;
          then a3,b2 // a3,b3 by A5,A13,A137,ANALMETR:63;
          then LIN a3,b2,b3 by ANALMETR:def 10;
          then LIN a39,b29,b39 by ANALMETR:40;
          then LIN b29,b39,a39 by AFF_1:6;
          hence contradiction by A11,A12,A14,A22,A28,AFF_1:25;
        end;
        then consider c39 be Element of the AffinStruct of X such that
A143:   c39 in A9 and
A144:   LIN b39,e19,c39 by A130,AFF_1:59;
A145:   b39,e19 // b39,c39 by A144,AFF_1:def 1;
        consider e19 be Element of the AffinStruct of X such that
A146:   c39,a39 // a29,e19 and
A147:   a29<> e19 by DIRAF:40;
        not a29,e19 // A9
        proof
A148:     c39<>o9
          proof
            assume c39=o9;
            then
A149:       a39,b29 // b39,o9 by A128,A129,A145,AFF_1:5;
            b39,o9 // b39,b29 by A9,A11,A12,A22,AFF_1:39,41;
            then a39,b29 // b39,b29 by A18,A149,AFF_1:5;
            then b29,b39 // b29,a39 by AFF_1:4;
            then LIN b29,b39,a39 by AFF_1:def 1;
            hence contradiction by A11,A12,A14,A22,A28,AFF_1:25;
          end;
          assume
A150:     a29,e19 // A9;
A151:     o9,a39 // o9,a39 by AFF_1:2;
          o9,c39 // A9 by A130,A131,A143,AFF_1:40,41;
          then a29,e19 // o9,c39 by A130,A150,AFF_1:31;
          then c39,a39 // o9,c39 by A146,A147,AFF_1:5;
          then c39,o9 // c39,a39 by AFF_1:4;
          then LIN c39,o9,a39 by AFF_1:def 1;
          then a39 in A9 by A130,A131,A143,A148,AFF_1:25;
          then A9 // M9 by A5,A8,A9,A14,A21,A130,A131,A151,AFF_1:38;
          then A // M by ANALMETR:46;
          hence contradiction by A23,A136,ANALMETR:52;
        end;
        then consider c29 be Element of the AffinStruct of X such that
A152:   c29 in A9 and
A153:   LIN a29,e19,c29 by A130,AFF_1:59;
        reconsider c3=c39 as Element of X;
        reconsider c2=c29 as Element of X;
        a29,e19 // a29,c29 by A153,AFF_1:def 1;
        then c39,a39 // a29,c29 by A146,A147,AFF_1:5;
        then
A154:   c3,a3 // a2,c2 by ANALMETR:36;
        then
A155:   a3,c3 // a2,c2 by ANALMETR:59;
        consider e19 be Element of the AffinStruct of X such that
A156:   c39,a39 // a19,e19 and
A157:   a19<> e19 by DIRAF:40;
        not a19,e19 // A9
        proof
A158:     c39<>o9
          proof
            assume c39=o9;
            then
A159:       a39,b29 // b39,o9 by A128,A129,A145,AFF_1:5;
            b39,o9 // b39,b29 by A9,A11,A12,A22,AFF_1:39,41;
            then a39,b29 // b39,b29 by A18,A159,AFF_1:5;
            then b29,b39 // b29,a39 by AFF_1:4;
            then LIN b29,b39,a39 by AFF_1:def 1;
            hence contradiction by A11,A12,A14,A22,A28,AFF_1:25;
          end;
          assume
A160:     a19,e19 // A9;
A161:     o9,a39 // o9,a39 by AFF_1:2;
          o9,c39 // A9 by A130,A131,A143,AFF_1:40,41;
          then a19,e19 // o9,c39 by A130,A160,AFF_1:31;
          then c39,a39 // o9,c39 by A156,A157,AFF_1:5;
          then c39,o9 // c39,a39 by AFF_1:4;
          then LIN c39,o9,a39 by AFF_1:def 1;
          then a39 in A9 by A130,A131,A143,A158,AFF_1:25;
          then A9 // M9 by A5,A8,A9,A14,A21,A130,A131,A161,AFF_1:38;
          then A // M by ANALMETR:46;
          hence contradiction by A23,A136,ANALMETR:52;
        end;
        then consider c19 be Element of the AffinStruct of X such that
A162:   c19 in A9 and
A163:   LIN a19,e19,c19 by A130,AFF_1:59;
        reconsider c1=c19 as Element of X;
        a19,e19 // a19,c19 by A163,AFF_1:def 1;
        then c39,a39 // a19,c19 by A156,A157,AFF_1:5;
        then
A164:   c3,a3 // a1,c1 by ANALMETR:36;
        then
A165:   a3,c3 // a1,c1 by ANALMETR:59;
A166:   a39,b29 // b39,c39 by A128,A129,A145,AFF_1:5;
A167:   o<>c3 & o<>c2 & o<>c1
        proof
A168:     now
            assume o=c3;
            then
A169:       a3,b2 // b3,o by A166,ANALMETR:36;
            b39,o9 // b29,b39 by A9,A11,A12,A22,AFF_1:39,41;
            then b3,o // b2,b3 by ANALMETR:36;
            then a3,b2 // b2,b3 by A18,A169,ANALMETR:60;
            then b2,b3 // b2,a3 by ANALMETR:59;
            then LIN b2,b3,a3 by ANALMETR:def 10;
            then LIN b29,b39,a39 by ANALMETR:40;
            hence contradiction by A11,A12,A14,A22,A28,AFF_1:25;
          end;
A170:     now
            a29,o9 // a29,a39 by A5,A7,A8,A21,AFF_1:39,41;
            then
A171:       a2,o // a2,a3 by ANALMETR:36;
            assume o=c2 or o=c1;
            then
A172:       a3,c3 // a2,o or a3,c3 // a1,o by A154,A164,ANALMETR:59;
            a19,o9 // a29,a39 by A5,A6,A7,A8,A21,AFF_1:39,41;
            then a1,o // a2,a3 by ANALMETR:36;
            then a3,c3 // a2, a3 by A15,A16,A172,A171,ANALMETR:60;
            then a3,a2 // a3,c3 by ANALMETR:59;
            then LIN a3,a2,c3 by ANALMETR:def 10;
            then LIN a39,a29,c39 by ANALMETR:40;
            then
A173:       c39 in M9 by A7,A8,A21,A26,AFF_1:25;
            o9,c39 // o9,c39 by AFF_1:2;
            then A9 // M9 by A5,A21,A130,A131,A143,A168,A173,AFF_1:38;
            then A // M by ANALMETR:46;
            hence contradiction by A23,A136,ANALMETR:52;
          end;
          assume o=c3 or o=c2 or o=c1;
          hence contradiction by A168,A170;
        end;
A174:   not c3 in N
        proof
A175:     o9,c39 // o9,c39 by AFF_1:2;
          assume c3 in N;
          then A9 // N9 by A9,A22,A130,A131,A143,A167,A175,AFF_1:38;
          then A // N by ANALMETR:46;
          hence contradiction by A136,ANALMETR:52;
        end;
A176:   not LIN a1,a2,c1
        proof
          assume LIN a1,a2,c1;
          then LIN a19,a29,c19 by ANALMETR:40;
          then
A177:     c19 in M9 by A6,A7,A21,A25,AFF_1:25;
          o9,c19 // o9,c19 by AFF_1:2;
          then A9 // M9 by A5,A21,A130,A131,A162,A167,A177,AFF_1:38;
          then A // M by ANALMETR:46;
          hence contradiction by A23,A136,ANALMETR:52;
        end;
A178:   not LIN a3,a1,c3 & not LIN a3,a2,c3
        proof
          assume LIN a3,a1,c3 or LIN a3,a2,c3;
          then LIN a39,a19,c39 or LIN a39,a29,c39 by ANALMETR:40;
          then
A179:     c39 in M9 or c39 in M9 by A6,A7,A8,A21,A26,A27,AFF_1:25;
          o9,c39 // o9,c39 by AFF_1:2;
          then A9 // M9 by A5,A21,A130,A131,A143,A167,A179,AFF_1:38;
          then A // M by ANALMETR:46;
          hence contradiction by A23,A136,ANALMETR:52;
        end;
A180:   not b2 in A
        proof
A181:     o9,b29 // o9,b29 by AFF_1:2;
          assume b2 in A;
          then A9 // N9 by A5,A9,A11,A13,A22,A130,A131,A181,AFF_1:38;
          then A // N by ANALMETR:46;
          hence contradiction by A136,ANALMETR:52;
        end;
A182:   not LIN c1,c2,b2
        proof
A183:     c19<>c29
          proof
A184:       a3<>c3
            proof
A185:         o9,c39 // o9,c39 by AFF_1:2;
              assume a3=c3;
              then A9 // M9 by A5,A8,A9,A14,A21,A130,A131,A143,A185,AFF_1:38;
              then A // M by ANALMETR:46;
              hence contradiction by A23,A136,ANALMETR:52;
            end;
            assume c19=c29;
            then a2,c1 // a1,c1 by A155,A165,A184,ANALMETR:60;
            then c1,a1 // c1,a2 by ANALMETR:59;
            then LIN c1,a1,a2 by ANALMETR:def 10;
            then LIN c19,a19,a29 by ANALMETR:40;
            then LIN a19,a29,c19 by AFF_1:6;
            then
A186:       c19 in M9 by A6,A7,A21,A25,AFF_1:25;
            o9,c19 // o9,c19 by AFF_1:2;
            then A9 // M9 by A5,A21,A130,A131,A162,A167,A186,AFF_1:38;
            then A // M by ANALMETR:46;
            hence contradiction by A23,A136,ANALMETR:52;
          end;
          assume LIN c1,c2,b2;
          then LIN c19,c29,b29 by ANALMETR:40;
          hence contradiction by A130,A152,A162,A180,A183,AFF_1:25;
        end;
        o9,c19 // o9,c29 by A130,A131,A152,A162,AFF_1:39,41;
        then LIN o9,c19,c29 by AFF_1:def 1;
        then
A187:   LIN o,c1,c2 by ANALMETR:40;
        a3<>c3
        proof
A188:     o9,a39 // o9,a39 by AFF_1:2;
          assume a3=c3;
          then A9 // M9 by A5,A8,A9,A14,A21,A130,A131,A143,A188,AFF_1:38;
          then A // M by ANALMETR:46;
          hence contradiction by A23,A136,ANALMETR:52;
        end;
        then a2,c2 // a1,c1 by A155,A165,ANALMETR:60;
        then
A189:   c1,a1 // c2,a2 by ANALMETR:59;
        o9,c39 // o9,c19 by A130,A131,A143,A162,AFF_1:39,41;
        then LIN o9,c39,c19 by AFF_1:def 1;
        then LIN o,c3,c1 by ANALMETR:40;
        then b3,c3 // b1,c1 by A2,A9,A14,A15,A17,A18,A20,A165,A167,A121,A178
,A124,A123,CONAFFM:def 1;
        then
A190:   c3,b3 // c1,b1 by ANALMETR:59;
        o9,c39 // o9,c29 by A130,A131,A143,A152,AFF_1:39,41;
        then LIN o9,c39,c29 by AFF_1:def 1;
        then LIN o,c3,c2 by ANALMETR:40;
        then b2,c3 // b1,c2 by A2,A5,A9,A13,A14,A16,A17,A19,A155,A167,A121,A178
,A122,A119,CONAFFM:def 1;
        then c3,b2 // c2,b1 by ANALMETR:59;
        then c1,b2 // c2,b3 by A1,A9,A10,A11,A12,A17,A18,A131,A136,A143,A152
,A162,A167,A190,A180,A174;
        hence thesis by A2,A5,A13,A15,A16,A18,A167,A118,A120,A187,A176,A182
,A189,CONAFFM:def 1;
      end;
      hence thesis by A40;
    end;
A191: now
A192: not LIN o9,a39,b39
      proof
        assume LIN o9,a39,b39;
        then LIN o9,b39,a39 by AFF_1:6;
        hence contradiction by A9,A12,A14,A18,A22,AFF_1:25;
      end;
      o9,b39 // o9,b19 by A9,A10,A12,A22,AFF_1:39,41;
      then
A193: LIN o9,b39,b19 by AFF_1:def 1;
      assume
A194: a1=a3;
      then a39,b39 // a39,b19 by A20,ANALMETR:36;
      hence thesis by A19,A194,A192,A193,AFF_1:14;
    end;
A195: now
      o9,b29 // o9,b39 by A9,A11,A12,A22,AFF_1:39,41;
      then
A196: LIN o9,b29,b39 by AFF_1:def 1;
      assume
A197: a1=a2;
      a1<>b1
      proof
        o9,a19 // o9,a39 by A5,A6,A8,A21,AFF_1:39,41;
        then
A198:   LIN o9,a19,a39 by AFF_1:def 1;
        assume a1=b1;
        hence contradiction by A9,A10,A14,A15,A22,A198,AFF_1:25;
      end;
      then a3,b2 // a3,b3 by A19,A20,A197,ANALMETR:60;
      then a39,b29 // a39,b39 by ANALMETR:36;
      then b29=b39 by A5,A8,A9,A13,A14,A21,A196,AFF_1:14,25;
      then a19,b29 // a29,b39 by A197,AFF_1:2;
      hence thesis by ANALMETR:36;
    end;
    now
      o9,b29 // o9,b19 by A9,A10,A11,A22,AFF_1:39,41;
      then
A199: LIN o9,b29,b19 by AFF_1:def 1;
      assume
A200: a2=a3;
      then a39,b29 // a39,b19 by A19,ANALMETR:36;
      then b29=b19 by A5,A8,A9,A13,A14,A21,A199,AFF_1:14,25;
      hence thesis by A20,A200,ANALMETR:59;
    end;
    hence thesis by A195,A191,A24;
  end;
  hence
  thesis by A1,A5,A6,A7,A8,A9,A10,A11,A12,A13,A14,A15,A16,A17,A18,A19,A20;
end;
