reserve OAS for OAffinSpace;
reserve a,a9,b,b9,c,c9,d,d1,d2,e1,e2,e3,e4,e5,e6,p,p9,q,r,x,y,z for Element of
  OAS;

theorem
  ex x st a,x // b,c & a,b // x,c
proof
A1: now
    consider e3 such that
A2: Mid b,c,e3 and
A3: c <>e3 by DIRAF:13;
A4: b,c // c,e3 by A2,DIRAF:def 3;
    assume
A5: not a,b,c are_collinear;
    then
A6: b<>c by DIRAF:31;
    then consider e4 such that
A7: a,c // c,e4 and
A8: a,b // e3,e4 by A4,ANALOAF:def 5;
A9: Mid a,c,e4 by A7,DIRAF:def 3;
    then Mid e4,c,a by DIRAF:9;
    then
A10: e4,c // c,a by DIRAF:def 3;
A11: e4<>c
    proof
      assume
A12:  e4=c;
      e3,c // c,b by A4,DIRAF:2;
      then a,b // c,b by A3,A8,A12,DIRAF:3;
      then b,a // b,c by DIRAF:2;
      then Mid b,a,c or Mid b,c,a by DIRAF:7;
      then b,a,c are_collinear or b,c,a are_collinear by DIRAF:28;
      hence contradiction by A5,DIRAF:30;
    end;
A13: not e4,c,e3 are_collinear
    proof
      b,c,e3 are_collinear  by A2,DIRAF:28;
      then
A14:  c,e3,b are_collinear  by DIRAF:30;
      assume
A15:  e4,c,e3 are_collinear ;
A16:  c,e3,c are_collinear  by DIRAF:31;
A17:  e4,c,c are_collinear  by DIRAF:31;
      e4,c,a are_collinear  by A9,DIRAF:9,28;
      then c,e3,a are_collinear  by A11,A15,A17,DIRAF:32;
      hence contradiction by A5,A3,A14,A16,DIRAF:32;
    end;
    b,c // c,e3 by A2,DIRAF:def 3;
    then
A18: c,e3 // b,c by DIRAF:2;
    consider e1 such that
A19: Mid c,a,e1 and
A20: a<>e1 by DIRAF:13;
A21: c,a // a,e1 by A19,DIRAF:def 3;
A22: a<>c by A5,DIRAF:31;
    then consider e2 such that
A23: b,a // a,e2 and
A24: b,c // e1,e2 by A21,ANALOAF:def 5;
    c,a // c,e1 by A21,ANALOAF:def 5;
    then e4,c // c,e1 by A22,A10,DIRAF:3;
    then
A25: Mid e4,c,e1 by DIRAF:def 3;
    then consider e5 such that
A26: Mid e4,e3,e5 and
A27: c,e3 // e1,e5 by A11,Th20;
A28: e4<>e3
    proof
      assume e4=e3;
      then Mid a,c,e3 by A7,DIRAF:def 3;
      then
A29:  e3,c,a are_collinear  by DIRAF:9,28;
A30:  e3,c,c are_collinear  by DIRAF:31;
      e3,c,b are_collinear  by A2,DIRAF:9,28;
      hence contradiction by A5,A3,A29,A30,DIRAF:32;
    end;
    then
A31: e4<>e5 by A26,DIRAF:8;
A32: e1<>e4 by A25,A11,DIRAF:8;
A33: not e1,e4,e3 are_collinear
    proof
      e4,c,e1 are_collinear  by A25,DIRAF:28;
      then
A34:  e4,e1,c are_collinear  by DIRAF:30;
      assume e1,e4,e3 are_collinear ;
      then
A35:  e4,e1,e3 are_collinear  by DIRAF:30;
      e4,e1,e4 are_collinear  by DIRAF:31;
      hence contradiction by A32,A13,A35,A34,DIRAF:32;
    end;
A36: not e1,e5,e4 are_collinear
    proof
      e4,e3,e5 are_collinear by A26,DIRAF:28;
      then
A37:  e5,e4,e3 are_collinear by DIRAF:30;
      assume e1,e5,e4 are_collinear;
      then
A38:  e5,e4,e1 are_collinear by DIRAF:30;
      e5,e4,e4 are_collinear by DIRAF:31;
      hence contradiction by A31,A33,A38,A37,DIRAF:32;
    end;
    then
A39: e1<>e5 by DIRAF:31;
    Mid e1,c,e4 by A25,DIRAF:9;
    then consider e6 such that
A40: Mid e1,e6,e5 and
A41: e5,e4 // e6,c by A36,Th21;
A42: c <>e1 by A19,A20,DIRAF:8;
A43: not c,e1,b are_collinear
    proof
      c,a,e1 are_collinear by A19,DIRAF:28;
      then
A44:  c,e1,a are_collinear by DIRAF:30;
A45:  c,e1,c are_collinear by DIRAF:31;
      assume c,e1,b are_collinear;
      hence contradiction by A5,A42,A44,A45,DIRAF:32;
    end;
A46: e5<>e3
    proof
      assume e5=e3;
      then e3,c // e3,e1 by A27,DIRAF:2;
      then Mid e3,c,e1 or Mid e3,e1,c by DIRAF:7;
      then e3,c,e1 are_collinear or e3,e1,c are_collinear by DIRAF:28;
      then
A47:  e3,c,e1 are_collinear by DIRAF:30;
A48:  e3,c,c are_collinear by DIRAF:31;
      e3,c,b are_collinear by A2,DIRAF:9,28;
      hence contradiction by A3,A43,A47,A48,DIRAF:32;
    end;
A49: e1<>e6
    proof
      Mid e5,e3,e4 by A26,DIRAF:9;
      then
A50:  e5,e3 // e3,e4 by DIRAF:def 3;
      then e5,e3 // e5,e4 by ANALOAF:def 5;
      then
A51:  e5,e4 // e3,e4 by A46,A50,ANALOAF:def 5;
      assume e1=e6;
      then e3,e4 // e1,c by A31,A41,A51,ANALOAF:def 5;
      then
A52:  a,b // e1,c by A8,A28,DIRAF:3;
      Mid e1,a,c by A19,DIRAF:9;
      then
A53:  e1,a // a,c by DIRAF:def 3;
      then e1,a // e1,c by ANALOAF:def 5;
      then e1,c // a,c by A20,A53,ANALOAF:def 5;
      then a,b // a,c by A42,A52,DIRAF:3;
      then a,b // b,c or a,c // c,b by DIRAF:6;
      then Mid a,b,c or Mid a,c,b by DIRAF:def 3;
      then a,b,c are_collinear or a,c,b are_collinear by DIRAF:28;
      hence contradiction by A5,DIRAF:30;
    end;
    consider x such that
A54: Mid c,x,e6 and
A55: e1,e6 // a,x by A19,Th19;
    e1,e6 // e6,e5 by A40,DIRAF:def 3;
    then e1,e6 // e1,e5 by ANALOAF:def 5;
    then e1,e5 // a,x by A55,A49,ANALOAF:def 5;
    then c,e3 // a,x by A27,A39,DIRAF:3;
    then
A56: a,x // b,c by A3,A18,ANALOAF:def 5;
A57: e6<>c
    proof
      assume e6=c;
      then x=c by A54,DIRAF:8;
      then c,a // c,b by A56,DIRAF:2;
      then Mid c,a,b or Mid c,b,a by DIRAF:7;
      then c,a,b are_collinear or c,b,a are_collinear by DIRAF:28;
      hence contradiction by A5,DIRAF:30;
    end;
A58: a<>e2
    proof
      assume
A59:  a=e2;
      e1,a // a,c by A21,DIRAF:2;
      then b,c // a,c by A20,A24,A59,DIRAF:3;
      then c,b // c,a by DIRAF:2;
      then Mid c,b,a or Mid c,a,b by DIRAF:7;
      then c,b,a are_collinear or c,a,b are_collinear by DIRAF:28;
      hence contradiction by A5,DIRAF:30;
    end;
A60: e6<>x
    proof
      assume e6=x;
      then e6,e1 // e6,a by A55,DIRAF:2;
      then Mid e6,e1,a or Mid e6,a,e1 by DIRAF:7;
      then e6,e1,a are_collinear or e6,a,e1 are_collinear by DIRAF:28;
      then
A61:  e6,e1,a are_collinear by DIRAF:30;
      e1,e6,e5 are_collinear by A40,DIRAF:28;
      then
A62:  e6,e1,e5 are_collinear by DIRAF:30;
      b,c // e1,e5 by A3,A4,A27,DIRAF:3;
      then e1,e2 // e1,e5 by A6,A24,ANALOAF:def 5;
      then Mid e1,e2,e5 or Mid e1,e5,e2 by DIRAF:7;
      then e1,e2,e5 are_collinear or e1,e5,e2 are_collinear by DIRAF:28;
      then
A63:  e1,e5,e2 are_collinear by DIRAF:30;
A64:  e1,e5,e1 are_collinear by DIRAF:31;
      Mid b,a,e2 by A23,DIRAF:def 3;
      then b,a,e2 are_collinear by DIRAF:28;
      then
A65:  a,e2,b are_collinear by DIRAF:30;
      c,a,e1 are_collinear by A19,DIRAF:28;
      then
A66:  a,e1,c are_collinear by DIRAF:30;
      e6,e1,e1 are_collinear by DIRAF:31;
      then e1,e5,a are_collinear by A49,A61,A62,DIRAF:32;
      then
A67:  a,e1,e2 are_collinear by A39,A63,A64,DIRAF:32;
A68:  a,e2,a are_collinear by DIRAF:31;
      a,e1,a are_collinear by DIRAF:31;
      then a,e2,c are_collinear by A20,A67,A66,DIRAF:32;
      hence contradiction by A5,A58,A65,A68,DIRAF:32;
    end;
    Mid e6,x,c by A54,DIRAF:9;
    then
A69: e6,x // x,c by DIRAF:def 3;
    then e6,x // e6,c by ANALOAF:def 5;
    then
A70: e6,c // x,c by A60,A69,ANALOAF:def 5;
    Mid e5,e3,e4 by A26,DIRAF:9;
    then
A71: e5,e3 // e3,e4 by DIRAF:def 3;
    then e5,e3 // e5,e4 by ANALOAF:def 5;
    then e3,e4 // e5,e4 by A46,A71,ANALOAF:def 5;
    then a,b // e5,e4 by A8,A28,DIRAF:3;
    then a,b // e6,c by A31,A41,DIRAF:3;
    then a,b // x,c by A57,A70,DIRAF:3;
    hence thesis by A56;
  end;
  now
A72: now
      assume Mid a,c,b;
      then a,c // c,b by DIRAF:def 3;
      then a,c // a,b by ANALOAF:def 5;
      hence a,a // b,c & a,b // a,c by DIRAF:2,4;
    end;
A73: now
      assume Mid b,a,c;
      then Mid c,a,b by DIRAF:9;
      then c,a // a,b by DIRAF:def 3;
      then c,a // c,b by ANALOAF:def 5;
      hence a,c // b,c & a,b // c,c by DIRAF:2,4;
    end;
A74: Mid a,b,c implies a,b // b,c & a,b // b,c by DIRAF:def 3;
    assume a,b,c are_collinear;
    hence thesis by A74,A73,A72,DIRAF:29;
  end;
  hence thesis by A1;
end;
