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 Th7:
  X is Desarguesian implies X is satisfying_major_Scherungssatz
proof
  assume
A1: X is Desarguesian;
  now
    let o,a1,a2,a3,a4,b1,b2,b3,b4,M,N;
    assume that
A2: M is being_line and
A3: N is being_line and
A4: o in M and
A5: o in N and
A6: a1 in M and
A7: a3 in M and
A8: b1 in M and
A9: b3 in M and
A10: a2 in N and
A11: a4 in N and
A12: b2 in N and
A13: b4 in N and
A14: not a4 in M and
A15: not a2 in M and
A16: not b2 in M and
    not b4 in M and
A17: not a1 in N and
A18: not a3 in N and
A19: not b1 in N and
    not b3 in N and
A20: a3,a2 // b3,b2 and
A21: a2,a1 // b2,b1 and
A22: a1,a4 // b1,b4;
A23: now
A24:  now
        assume ex x,y,z st LIN x,y,z & x<>y & x<>z & y<>z;
        then consider d such that
A25:    LIN a1,a2,d and
A26:    a1<>d and
A27:    a2<>d by TRANSLAC:1;
        LIN o,d,d by AFF_1:7;
        then consider Y such that
A28:    Y is being_line and
A29:    o in Y and
A30:    d in Y and
        d in Y by AFF_1:21;
        a1,a2 // a1,d by A25,AFF_1:def 1;
        then a2,a1 // a1,d by AFF_1:4;
        then
A31:    b2,b1 // a1,d by A6,A15,A21,AFF_1:5;
A32:    N<>M & N<>Y & M<>Y
        proof
A33:      now
A34:        LIN a1,d,a2 by A25,AFF_1:6;
            assume
A35:        N=Y or M=Y;
            LIN a2,d,a1 by A25,AFF_1:6;
            hence contradiction by A2,A3,A6,A10,A15,A17,A26,A27,A30,A35,A34,
AFF_1:25;
          end;
          assume not thesis;
          hence contradiction by A11,A14,A33;
        end;
A36:    ex d1 st LIN b1,b2,d1 & d1 in Y
        proof
          not b1,b2 // o,d
          proof
A37:        o<>d
            proof
              assume o=d;
              then LIN o,a1,a2 by A25,AFF_1:6;
              hence contradiction by A2,A4,A5,A6,A15,A17,AFF_1:25;
            end;
            assume b1,b2 // o,d;
            then b2,b1 // o,d by AFF_1:4;
            then
A38:        a2,a1 // o,d by A8,A16,A21,AFF_1:5;
            a1,a2 // a1,d by A25,AFF_1:def 1;
            then a2,a1 // a1,d by AFF_1:4;
            then a1,d // o,d by A6,A15,A38,AFF_1:5;
            then d,o // d,a1 by AFF_1:4;
            then LIN d,o,a1 by AFF_1:def 1;
            then LIN o,d,a1 by AFF_1:6;
            then o,d // o,a1 by AFF_1:def 1;
            then
A39:        Y // M by A2,A4,A5,A6,A17,A28,A29,A30,A37,AFF_1:38;
            LIN a2,a1,d by A25,AFF_1:6;
            then a2,a1 // a2,d by AFF_1:def 1;
            then a2,d // o,d by A6,A15,A38,AFF_1:5;
            then d,a2 // d,o by AFF_1:4;
            then LIN d,a2,o by AFF_1:def 1;
            then LIN o,d,a2 by AFF_1:6;
            then o,d // o,a2 by AFF_1:def 1;
            then Y // N by A3,A4,A5,A10,A15,A28,A29,A30,A37,AFF_1:38;
            then M // N by A39,AFF_1:44;
            hence contradiction by A4,A5,A11,A14,AFF_1:45;
          end;
          then consider d1 such that
A40:      LIN b1,b2,d1 and
A41:      LIN o,d,d1 by AFF_1:60;
          take d1;
          o<>d
          proof
            assume o=d;
            then LIN o,a1,a2 by A25,AFF_1:6;
            hence contradiction by A2,A4,A5,A6,A15,A17,AFF_1:25;
          end;
          hence thesis by A28,A29,A30,A40,A41,AFF_1:25;
        end;
        LIN a2,a1,d by A25,AFF_1:6;
        then a2,a1 // a2,d by AFF_1:def 1;
        then
A42:    b2,b1 // a2,d by A6,A15,A21,AFF_1:5;
A43:    o<>d
        proof
          assume o=d;
          then LIN o,a1,a2 by A25,AFF_1:6;
          hence contradiction by A2,A4,A5,A6,A15,A17,AFF_1:25;
        end;
        consider d1 such that
A44:    LIN b1,b2,d1 and
A45:    d1 in Y by A36;
        b1,b2 // b1,d1 by A44,AFF_1:def 1;
        then b2,b1 // b1,d1 by AFF_1:4;
        then a1,d // b1,d1 by A8,A16,A31,AFF_1:5;
        then
A46:    d,a4 // d1,b4 by A1,A2,A3,A4,A5,A6,A8,A11,A13,A14,A17,A22,A28,A29,A30
,A45,A43,A32,AFF_2:def 4;
        LIN b2,b1,d1 by A44,AFF_1:6;
        then b2,b1 // b2,d1 by AFF_1:def 1;
        then
A47:    a2,d // b2,d1 by A8,A16,A42,AFF_1:5;
        a2,a3 // b2,b3 by A20,AFF_1:4;
        then d,a3 // d1,b3 by A1,A2,A3,A4,A5,A7,A9,A10,A12,A15,A18,A28,A29,A30
,A45,A43,A32,A47,AFF_2:def 4;
        hence
        a3,a4 // b3,b4 by A1,A2,A3,A4,A5,A7,A9,A11,A13,A14,A18,A28,A29,A30,A45
,A43,A32,A46,AFF_2:def 4;
      end;
      assume that
A48:  a1<>a3 and
A49:  a2<>a4;
      now
        assume
A50:    not ex x,y,z st LIN x,y,z & x<>y & x<>z & y<>z;
A51:    now
          assume
A52:      a1=b1;
A53:      a2<>b4
          proof
            assume a2=b4;
            then LIN a1,a4,a2 by A22,A52,AFF_1:def 1;
            then LIN a2,a4,a1 by AFF_1:6;
            hence contradiction by A3,A10,A11,A17,A49,AFF_1:25;
          end;
A54:      a1<>b3
          proof
            assume a1=b3;
            then b2,b1 // a2,a3 by A20,A52,AFF_1:4;
            then
A55:        a2,a1 // a2,a3 by A8,A16,A21,AFF_1:5;
            LIN o,a1,a3 by A2,A4,A6,A7,AFF_1:21;
            hence contradiction by A3,A4,A5,A10,A15,A17,A48,A55,AFF_1:14,25;
          end;
          LIN a2,a4,b4 by A3,A10,A11,A13,AFF_1:21;
          then
A56:      a2=b4 or a4=b4 by A49,A50;
          LIN a1,a3,b3 by A2,A6,A7,A9,AFF_1:21;
          then a3=b3 by A48,A50,A54;
          hence a3,a4 // b3,b4 by A56,A53,AFF_1:2;
        end;
A57:    now
          assume
A58:      a3=b1;
A59:      a2<>b2
          proof
            assume a2=b2;
            then LIN a2,a1,a3 by A21,A58,AFF_1:def 1;
            then LIN a1,a3,a2 by AFF_1:6;
            hence contradiction by A2,A6,A7,A15,A48,AFF_1:25;
          end;
A60:      a3<>b3
          proof
            assume a3=b3;
            then a3,a2 // b2,b1 by A20,A58,AFF_1:4;
            then a3,a2 // a2,a1 by A8,A16,A21,AFF_1:5;
            then a2,a3 // a2,a1 by AFF_1:4;
            then LIN a2,a3,a1 by AFF_1:def 1;
            then LIN a1,a3,a2 by AFF_1:6;
            hence contradiction by A2,A6,A7,A15,A48,AFF_1:25;
          end;
A61:      a4<>b4
          proof
            assume a4=b4;
            then a4,a1 // a4,a3 by A22,A58,AFF_1:4;
            then LIN a4,a1,a3 by AFF_1:def 1;
            then LIN a1,a3,a4 by AFF_1:6;
            hence contradiction by A2,A6,A7,A14,A48,AFF_1:25;
          end;
          LIN a2,a4,b4 by A3,A10,A11,A13,AFF_1:21;
          then
A62:      a2=b4 by A49,A50,A61;
          LIN a2,a4,b2 by A3,A10,A11,A12,AFF_1:21;
          then a4=b2 by A49,A50,A59;
          then
A63:      a3,a4 // b2,b1 by A58,AFF_1:2;
          LIN a1,a3,b3 by A2,A6,A7,A9,AFF_1:21;
          then a1=b3 by A48,A50,A60;
          then a3,a4 // b4,b3 by A8,A16,A21,A62,A63,AFF_1:5;
          hence a3,a4 // b3,b4 by AFF_1:4;
        end;
        LIN a1,a3,b1 by A2,A6,A7,A8,AFF_1:21;
        hence a3,a4 // b3,b4 by A48,A50,A51,A57;
      end;
      hence a3,a4 // b3,b4 by A24;
    end;
    now
A64:  now
        o,b2 // o,b4 by A3,A5,A12,A13,AFF_1:39,41;
        then
A65:    LIN o,b2,b4 by AFF_1:def 1;
        assume
A66:    a2=a4;
        a1,a2 // b1,b2 by A21,AFF_1:4;
        then b1,b2 // b1,b4 by A6,A14,A22,A66,AFF_1:5;
        hence a3,a4 // b3,b4 by A2,A4,A5,A8,A16,A19,A20,A66,A65,AFF_1:14,25;
      end;
A67:  now
        o,b1 // o,b3 by A2,A4,A8,A9,AFF_1:39,41;
        then
A68:    LIN o,b1,b3 by AFF_1:def 1;
        assume
A69:    a1=a3;
        then a2,a1 // b2,b3 by A20,AFF_1:4;
        then b2,b1 // b2,b3 by A6,A15,A21,AFF_1:5;
        hence a3,a4 // b3,b4 by A3,A4,A5,A12,A16,A19,A22,A69,A68,AFF_1:14,25;
      end;
      assume a1=a3 or a2=a4;
      hence a3,a4 // b3,b4 by A67,A64;
    end;
    hence a3,a4 // b3,b4 by A23;
  end;
  hence thesis;
end;
