reserve SAS for Semi_Affine_Space;
reserve a,a9,a1,a2,a3,a4,b,b9,c,c9,d,d9,d1,d2,o,p,p1,p2,q,r,r1,r2,s,x, y,t,z
  for Element of SAS;

theorem Th27:
  not a,b,c are_collinear & a,b // c,d implies not a,b,d are_collinear
proof
  assume that
A1: not a,b,c are_collinear and
A2: a,b // c,d;
  now
    assume that
A3: c <>d and
A4: a,b,d are_collinear;
    a,b // a,d by A4;
    then
A5: a,b // d,a by Th6;
A6: a,b // d,c by A2,Th6;
A7: a,c // c,a by Def1;
A8: not a,b // a,c by A1;
    then a<>b by Th3;
    then
A9: d,c // d,a by A6,A5,Def1;
    c <>a by A8,Def1;
    then not c,d // c,a by A2,A3,A8,A7,Th15;
    hence contradiction by A9,Th7;
  end;
  hence thesis by A1;
end;
