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 Th6:
  X is translational implies X is satisfying_minor_Scherungssatz
proof
  assume
A1: X is translational;
  now
    let a1,a2,a3,a4,b1,b2,b3,b4,M,N;
    assume that
A2: M // N and
A3: a1 in M and
A4: a3 in M and
A5: b1 in M and
A6: b3 in M and
A7: a2 in N and
A8: a4 in N and
A9: b2 in N and
A10: b4 in N and
A11: not a4 in M and
A12: not a2 in M and
A13: not b2 in M and
    not b4 in M and
A14: not a1 in N and
A15: not a3 in N and
A16: not b1 in N and
    not b3 in N and
A17: a3,a2 // b3,b2 and
A18: a2,a1 // b2,b1 and
A19: a1,a4 // b1,b4;
A20: M is being_line by A2,AFF_1:36;
A21: N is being_line by A2,AFF_1:36;
A22: now
      assume that
A23:  a1<>a3 and
A24:  a2<>a4;
A25:  now
        assume ex x,y,z st LIN x,y,z & x<>y & x<>z & y<>z;
        then consider d such that
A26:    LIN a1,a2,d and
A27:    a1<>d and
A28:    a2<>d by TRANSLAC:1;
A29:    ex Y st Y is being_line & d in Y & Y // M
        proof
          consider Y such that
A30:      d in Y and
A31:      M // Y by A20,AFF_1:49;
          take Y;
          thus thesis by A30,A31,AFF_1:36;
        end;
        LIN a2,a1,d by A26,AFF_1:6;
        then a2,a1 // a2,d by AFF_1:def 1;
        then
A32:    b2,b1 // a2,d by A3,A12,A18,AFF_1:5;
A33:    a2,a3 // b2,b3 by A17,AFF_1:4;
        consider Y such that
A34:    Y is being_line and
A35:    d in Y and
A36:    Y // M by A29;
A37:    Y // N by A2,A36,AFF_1:44;
A38:    N<>M & N<>Y & Y<>M
        proof
A39:      now
            LIN a2,d,a1 by A26,AFF_1:6;
            then
A40:        a2,d // a2,a1 by AFF_1:def 1;
            assume N=Y;
            then a2,d // a2,a4 by A7,A8,A34,A35,AFF_1:39,41;
            then a2,a4 // a2,a1 by A28,A40,AFF_1:5;
            then LIN a2,a4,a1 by AFF_1:def 1;
            hence contradiction by A7,A8,A14,A21,A24,AFF_1:25;
          end;
A41:      now
            assume Y=M;
            then
A42:        a1,d // a1,a3 by A3,A4,A35,A36,AFF_1:39;
            a1,a2 // a1,d by A26,AFF_1:def 1;
            then a1,a3 // a1,a2 by A27,A42,AFF_1:5;
            then LIN a1,a3,a2 by AFF_1:def 1;
            hence contradiction by A3,A4,A12,A20,A23,AFF_1:25;
          end;
          assume not thesis;
          hence contradiction by A8,A11,A39,A41;
        end;
A43:    Y // Y by A34,AFF_1:41;
        ex d1 st d1 in Y & b1,b2 // b1,d1
        proof
          consider e1 such that
A44:      d<>e1 and
A45:      e1 in Y by A34,AFF_1:20;
          consider e2 such that
A46:      b1,b2 // b1,e2 and
A47:      b1<>e2 by DIRAF:40;
          not d,e1 // b1,e2
          proof
            assume d,e1 // b1,e2;
            then d,e1 // b1,b2 by A46,A47,AFF_1:5;
            then
A48:        b1,b2 // Y by A35,A43,A44,A45,AFF_1:32,40;
            a1,a3 // Y by A3,A4,A36,AFF_1:40;
            then a1,a3 // b1,b2 by A34,A48,AFF_1:31;
            then a1,a3 // b2,b1 by AFF_1:4;
            then a1,a3 // a2,a1 by A5,A13,A18,AFF_1:5;
            then a1,a3 // a1,a2 by AFF_1:4;
            then LIN a1,a3,a2 by AFF_1:def 1;
            hence contradiction by A3,A4,A12,A20,A23,AFF_1:25;
          end;
          then consider d1 such that
A49:      LIN d,e1,d1 and
A50:      LIN b1,e2,d1 by AFF_1:60;
          take d1;
          b1,e2 // b1,d1 by A50,AFF_1:def 1;
          hence thesis by A34,A35,A44,A45,A46,A47,A49,AFF_1:5,25;
        end;
        then consider d1 such that
A51:    d1 in Y and
A52:    b1,b2 // b1,d1;
        a1,a2 // a1,d by A26,AFF_1:def 1;
        then a2,a1 // a1,d by AFF_1:4;
        then b2,b1 // a1,d by A3,A12,A18,AFF_1:5;
        then b1,b2 // a1,d by AFF_1:4;
        then a1,d // b1,d1 by A5,A13,A52,AFF_1:5;
        then
A53:    d,a4 // d1,b4 by A1,A2,A3,A5,A8,A10,A19,A20,A21,A34,A35,A36,A51,A38,
AFF_2:def 11;
        LIN b1,b2,d1 by A52,AFF_1:def 1;
        then LIN b2,b1,d1 by AFF_1:6;
        then b2,b1 // b2,d1 by AFF_1:def 1;
        then
A54:    a2,d // b2,d1 by A5,A13,A32,AFF_1:5;
        N // Y by A2,A36,AFF_1:44;
        then d,a3 // d1,b3 by A1,A2,A4,A6,A7,A9,A20,A21,A34,A35,A51,A38,A54,A33
,AFF_2:def 11;
        hence a3,a4 // b3,b4 by A1,A4,A6,A8,A10,A20,A21,A34,A35,A36,A51,A38,A53
,A37,AFF_2:def 11;
      end;
      now
        assume
A55:    not ex x,y,z st LIN x,y,z & x<>y & x<>z & y<>z;
A56:    now
          assume
A57:      a1=b1;
          then
A58:      LIN a1,a4,b4 by A19,AFF_1:def 1;
          a1,a2 // a1,b2 by A18,A57,AFF_1:4;
          then
A59:      LIN a1,a2,b2 by AFF_1:def 1;
          a2,b2 // M by A2,A7,A9,AFF_1:40;
          then a2=b2 by A3,A12,A59,AFF_1:46;
          then a2,a3 // a2,b3 by A17,AFF_1:4;
          then
A60:      LIN a2,a3,b3 by AFF_1:def 1;
          a3,b3 // N by A2,A4,A6,AFF_1:40;
          then
A61:      a3=b3 by A7,A15,A60,AFF_1:46;
          a4,b4 // M by A2,A8,A10,AFF_1:40;
          then a4=b4 by A3,A11,A58,AFF_1:46;
          hence a3,a4 // b3,b4 by A61,AFF_1:2;
        end;
A62:    now
          assume
A63:      a3=b1;
A64:      now
A65:        a4<>b4
            proof
              assume a4=b4;
              then a4,a1 // a4,b1 by A19,AFF_1:4;
              then
A66:          LIN a4,a1,b1 by AFF_1:def 1;
              a1,b1 // N by A2,A3,A5,AFF_1:40;
              hence contradiction by A8,A14,A23,A63,A66,AFF_1:46;
            end;
            a2,a4 // a2,b4 by A7,A8,A10,A21,AFF_1:39,41;
            then LIN a2,a4,b4 by AFF_1:def 1;
            then
A67:        a2=b4 by A24,A55,A65;
            assume
A68:        a4=b2;
A69:        a3<>b3
            proof
              assume a3=b3;
              then
A70:          LIN a3,a2,b2 by A17,AFF_1:def 1;
              a2,b2 // M by A2,A7,A9,AFF_1:40;
              hence contradiction by A4,A12,A24,A68,A70,AFF_1:46;
            end;
            a1,a3 // a1,b3 by A3,A4,A6,A20,AFF_1:39,41;
            then LIN a1,a3,b3 by AFF_1:def 1;
            then
A71:        a1=b3 by A23,A55,A69;
            a3,a4 // b2,b1 by A63,A68,AFF_1:2;
            then a3,a4 // a2,a1 by A5,A13,A18,AFF_1:5;
            hence a3,a4 // b3,b4 by A71,A67,AFF_1:4;
          end;
A72:      now
            assume a2=b2;
            then
A73:        LIN a2,a1,b1 by A18,AFF_1:def 1;
            a1,b1 // N by A2,A3,A5,AFF_1:40;
            hence a3,a4 // b3,b4 by A7,A14,A56,A73,AFF_1:46;
          end;
          a2,a4 // a2,b2 by A7,A8,A9,A21,AFF_1:39,41;
          then LIN a2,a4,b2 by AFF_1:def 1;
          hence a3,a4 // b3,b4 by A24,A55,A64,A72;
        end;
        a1,a3 // a1,b1 by A3,A4,A5,A20,AFF_1:39,41;
        then LIN a1,a3,b1 by AFF_1:def 1;
        hence a3,a4 // b3,b4 by A23,A55,A56,A62;
      end;
      hence a3,a4 // b3,b4 by A25;
    end;
    now
A74:  now
        assume
A75:    a1=a3;
        b1=b3
        proof
          LIN a2,a1,a3 by A75,AFF_1:7;
          then a2,a1 // a2,a3 by AFF_1:def 1;
          then b2,b1 // a2,a3 by A3,A12,A18,AFF_1:5;
          then b2,b1 // a3,a2 by AFF_1:4;
          then b2,b1 // b3,b2 by A4,A12,A17,AFF_1:5;
          then b2,b1 // b2,b3 by AFF_1:4;
          then
A76:      LIN b2,b1,b3 by AFF_1:def 1;
          assume
A77:      b1<>b3;
          b1,b3 // N by A2,A5,A6,AFF_1:40;
          hence contradiction by A9,A16,A77,A76,AFF_1:46;
        end;
        hence a3,a4 // b3,b4 by A19,A75;
      end;
A78:  now
        assume
A79:    a2=a4;
        then LIN a1,a4,a2 by AFF_1:7;
        then a1,a4 // a1,a2 by AFF_1:def 1;
        then b1,b4 // a1,a2 by A3,A11,A19,AFF_1:5;
        then b1,b4 // a2,a1 by AFF_1:4;
        then b1,b4 // b2,b1 by A3,A12,A18,AFF_1:5;
        then b1,b2 // b1,b4 by AFF_1:4;
        then
A80:    LIN b1,b2,b4 by AFF_1:def 1;
        b2,b4 // M by A2,A9,A10,AFF_1:40;
        hence a3,a4 // b3,b4 by A5,A13,A17,A79,A80,AFF_1:46;
      end;
      assume a1=a3 or a2=a4;
      hence a3,a4 // b3,b4 by A74,A78;
    end;
    hence a3,a4 // b3,b4 by A22;
  end;
  hence thesis;
end;
