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 Th23:
  a,b // c,d & not a,b,c are_collinear implies ex x st Mid a,x,d & Mid b,x,c
proof
  assume that
A1: a,b // c,d and
A2: not a,b,c are_collinear;
A3: now
    consider e1 such that
A4: a,b // d,e1 and
A5: a,d // b,e1 and
A6: b<>e1 by ANALOAF:def 5;
A7: a<>b by A2,DIRAF:31;
    then c,d // d,e1 by A1,A4,ANALOAF:def 5;
    then
A8: Mid c,d,e1 by DIRAF:def 3;
A9: a<>d
    proof
      assume a=d;
      then b,a // a,c by A1,DIRAF:2;
      then Mid b,a,c by DIRAF:def 3;
      then b,a,c are_collinear by DIRAF:28;
      hence contradiction by A2,DIRAF:30;
    end;
A10: not a,b,d are_collinear
    proof
A11:  now
        assume a,b // a,d;
        then c,d // a,d by A1,A7,ANALOAF:def 5;
        then d,c // d,a by DIRAF:2;
        then d,c '||' d,a by DIRAF:def 4;
        hence d,c,a are_collinear by DIRAF:def 5;
      end;
A12:  d,a,a are_collinear by DIRAF:31;
A13:  now
        assume a,b // d,a;
        then c,d // d,a by A1,A7,ANALOAF:def 5;
        then Mid c,d,a by DIRAF:def 3;
        then c,d,a are_collinear by DIRAF:28;
        hence d,c,a are_collinear by DIRAF:30;
      end;
      assume
A14:  a,b,d are_collinear;
      then
A15:  d,a,b are_collinear by DIRAF:30;
      a,b '||' a,d by A14,DIRAF:def 5;
      then d,a,c are_collinear by A11,A13,DIRAF:30,def 4;
      hence contradiction by A2,A9,A12,A15,DIRAF:32;
    end;
    consider e2 such that
A16: c,d // b,e2 and
A17: c,b // d,e2 and
A18: d<>e2 by ANALOAF:def 5;
    assume
A19: c <>d;
    then a,b // b,e2 by A1,A16,DIRAF:3;
    then
A20: Mid a,b,e2 by DIRAF:def 3;
A21: not c,d,b are_collinear
    proof
A22:  now
        assume c,d // c,b;
        then a,b // c,b by A1,A19,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 A2,DIRAF:30;
      end;
      assume c,d,b are_collinear;
      then c,d '||' c,b by DIRAF:def 5;
      then c,d // c,b or c,d // b,c by DIRAF:def 4;
      then a,b // b,c by A1,A19,A22,DIRAF:3;
      then Mid a,b,c by DIRAF:def 3;
      hence contradiction by A2,DIRAF:28;
    end;
A23: b,c,c are_collinear by DIRAF:31;
A24: d,a,a are_collinear by DIRAF:31;
A25: c <>e1
    proof
      assume c =e1;
      then c,d // d,c by A1,A4,A7,ANALOAF:def 5;
      hence contradiction by A19,ANALOAF:def 5;
    end;
    not c,b,e1 are_collinear
    proof
      c,d,e1 are_collinear by A8,DIRAF:28;
      then
A26:  c,e1,d are_collinear by DIRAF:30;
      assume c,b,e1 are_collinear;
      then
A27:  c,e1,b are_collinear by DIRAF:30;
      c,e1,c are_collinear by DIRAF:31;
      hence contradiction by A25,A21,A27,A26,DIRAF:32;
    end;
    then consider x such that
A28: Mid c,x,b and
A29: b,e1 // x,d by A8,Th21;
A30: Mid b,x,c by A28,DIRAF:9;
    a,d // x,d by A5,A6,A29,DIRAF:3;
    then d,a // d,x by DIRAF:2;
    then Mid d,a,x or Mid d,x,a by DIRAF:7;
    then d,a,x are_collinear or d,x,a are_collinear by DIRAF:28;
    then
A31: d,a,x are_collinear by DIRAF:30;
A32: b<>c by A2,DIRAF:31;
A33: a<>e2
    proof
      assume a=e2;
      then a,b // b,a by A1,A19,A16,DIRAF:3;
      hence contradiction by A7,ANALOAF:def 5;
    end;
    not a,d,e2 are_collinear
    proof
      a,b,e2 are_collinear by A20,DIRAF:28;
      then
A34:  a,e2,b are_collinear by DIRAF:30;
      assume a,d,e2 are_collinear;
      then
A35:  a,e2,d are_collinear by DIRAF:30;
      a,e2,a are_collinear by DIRAF:31;
      hence contradiction by A33,A10,A35,A34,DIRAF:32;
    end;
    then consider y such that
A36: Mid a,y,d and
A37: d,e2 // y,b by A20,Th21;
A38: b,c,b are_collinear by DIRAF:31;
    c,b // y,b by A17,A18,A37,DIRAF:3;
    then b,c // b,y by DIRAF:2;
    then Mid b,c,y or Mid b,y,c by DIRAF:7;
    then b,c,y are_collinear or b,y,c are_collinear by DIRAF:28;
    then
A39: b,c,y are_collinear by DIRAF:30;
A40: c,x,b are_collinear by A28,DIRAF:28;
    then b,c,x are_collinear by DIRAF:30;
    then
A41: x,y,c are_collinear by A32,A39,A23,DIRAF:32;
    a,y,d are_collinear by A36,DIRAF:28;
    then d,a,y are_collinear by DIRAF:30;
    then
A42: x,y,a are_collinear by A9,A31,A24,DIRAF:32;
    b,c,x are_collinear by A40,DIRAF:30;
    then x,y,b are_collinear by A32,A39,A38,DIRAF:32;
    then Mid a,x,d by A2,A36,A42,A41,DIRAF:32;
    hence thesis by A30;
  end;
  now
    assume
A43: c =d;
    take x=c;
    thus Mid a,x,d & Mid b,x,c by A43,DIRAF:10;
  end;
  hence thesis by A3;
end;
