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 Th10:
  not a,b,a9 are_collinear & a,a9 '||' b,b9 & a,a9 '||' c,c9 & Mid a,b,c &
  a9,b9,c9 are_collinear implies Mid a9,b9,c9
proof
  assume that
A1: not a,b,a9 are_collinear and
A2: a,a9 '||' b,b9 and
A3: a,a9 '||' c,c9 and
A4: Mid a,b,c and
A5: a9,b9,c9 are_collinear;
A6: a,b,c are_collinear by A4,DIRAF:28;
A7: b9,c9,a9 are_collinear by A5,DIRAF:30;
A8: c9,b9,a9 are_collinear by A5,DIRAF:30;
A9: a<>a9 by A1,DIRAF:31;
  then
A10: b,b9 '||' c,c9 by A2,A3,DIRAF:23;
A11: a<>b by A1,DIRAF:31;
  then
A12: a<>c by A4,DIRAF:8;
A13: now
    assume that
A14: b9<>c9 and
    a9<>b9 and
A15: b<>b9;
A16: not b,b9,a9 are_collinear
    proof
A17:  b,b9,b are_collinear by DIRAF:31;
      assume
A18:  b,b9,a9 are_collinear;
      b,b9 '||' a9,a by A2,DIRAF:22;
      then b,b9,a are_collinear by A15,A18,DIRAF:33;
      hence contradiction by A1,A15,A18,A17,DIRAF:32;
    end;
A19: now
      a,b '||' a,c by A6,DIRAF:def 5;
      then c,a '||' a,b by DIRAF:22;
      then consider x such that
A20:  c9,a '||' a,x and
A21:  c9,c '||' b,x by A12,DIRAF:27;
      a,c9 '||' a,x by A20,DIRAF:22;
      then
A22:  a,c9,x are_collinear by DIRAF:def 5;
      assume
A23:  c <>c9;
A24:  x<>b
      proof
        assume x=b;
        then
A25:    a,b,c9 are_collinear by A22,DIRAF:30;
        a,b,b are_collinear by DIRAF:31;
        then
A26:    c,c9,b are_collinear by A11,A6,A25,DIRAF:32;
        a,b,a are_collinear by DIRAF:31;
        then c,c9,a are_collinear by A11,A6,A25,DIRAF:32;
        then c,c9 '||' a,b by A26,DIRAF:34;
        then b,b9 '||' a,b by A10,A23,DIRAF:23;
        then a,a9 '||' a,b by A2,A15,DIRAF:23;
        then a,a9,b are_collinear by DIRAF:def 5;
        hence contradiction by A1,DIRAF:30;
      end;
      c,c9 '||' b,x by A21,DIRAF:22;
      then b,b9 '||' b,x by A10,A23,DIRAF:23;
      then
A27:  b,b9,x are_collinear by DIRAF:def 5;
      then b,x,b9 are_collinear by DIRAF:30;
      then b,x '||' b,b9 by DIRAF:def 5;
      then b,x '||' c,c9 by A10,A15,DIRAF:23;
      then
A28:  x,b '||' c,c9 by DIRAF:22;
A29:  x<>b9
      proof
        assume x=b9;
        then
A30:    b9,c9,a are_collinear by A22,DIRAF:30;
A31:    a,a9 '||' b9,b by A2,DIRAF:22;
        b9,c9,b9 are_collinear by DIRAF:31;
        then a,a9,b9 are_collinear by A7,A14,A30,DIRAF:32;
        then a,a9,b are_collinear by A9,A31,DIRAF:33;
        hence contradiction by A1,DIRAF:30;
      end;
A32:  not c9,b9,x are_collinear
      proof
        assume c9,b9,x are_collinear;
        then
A33:    b9,x,c9 are_collinear by DIRAF:30;
A34:    b9,x,b9 are_collinear by DIRAF:31;
A35:    c9,b9,b9 are_collinear by DIRAF:31;
        b9,x,b are_collinear by A27,DIRAF:30;
        then c9,b9,b are_collinear by A29,A33,A34,DIRAF:32;
        hence contradiction by A8,A14,A16,A35,DIRAF:32;
      end;
A36:  x,b,b9 are_collinear by A27,DIRAF:30;
A37:  not a,x,b are_collinear
      proof
        assume a,x,b are_collinear;
        then
A38:    x,b,a are_collinear by DIRAF:30;
A39:    b,b9 '||' a,a9 by A2,DIRAF:22;
        x,b,b are_collinear by DIRAF:31;
        then b,b9,a are_collinear by A36,A24,A38,DIRAF:32;
        hence contradiction by A15,A16,A39,DIRAF:33;
      end;
      b9,b,x are_collinear by A27,DIRAF:30;
      then b9,b '||' b9,x by DIRAF:def 5;
      then b,b9 '||' b9,x by DIRAF:22;
      then
A40:  b9,x '||' a,a9 by A2,A15,DIRAF:23;
      a,x,c9 are_collinear by A22,DIRAF:30;
      then Mid a,x,c9 by A4,A28,A37,Th8;
      then Mid c9,x,a by DIRAF:9;
      then Mid c9,b9,a9 by A8,A40,A32,Th8;
      hence thesis by DIRAF:9;
    end;
    c =c9 implies thesis
    proof
A41:  not c9,b9,b are_collinear
      proof
A42:    c9,b9,b9 are_collinear by DIRAF:31;
        assume c9,b9,b are_collinear;
        hence contradiction by A8,A14,A16,A42,DIRAF:32;
      end;
      assume c =c9;
      then
A43:  Mid c9,b,a by A4,DIRAF:9;
      b9,b '||' a,a9 by A2,DIRAF:22;
      then Mid c9,b9,a9 by A8,A43,A41,Th8;
      hence thesis by DIRAF:9;
    end;
    hence thesis by A19;
  end;
  b=b9 implies thesis
  proof
A44: a,a9 '||' c9,c by A3,DIRAF:22;
A45: b9,a9,c9 are_collinear by A5,DIRAF:30;
    assume
A46: b=b9;
    then not b9,a,a9 are_collinear by A1,DIRAF:30;
    hence thesis by A4,A46,A45,A44,Th6;
  end;
  hence thesis by A13,DIRAF:10;
end;
