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 Th32:
  a<>b & a,b,c are_collinear & a,b // c,d implies a,c // b,d
proof
  assume that
A1: a<>b and
A2: a,b,c are_collinear and
A3: a,b // c,d;
  now
A4: a,b // a,c by A2;
    then a,b // c,b by Th7;
    then c,b // c,d by A1,A3,Def1;
    then
A5: b,c // b,d by Th7;
    assume
A6: b<>c;
    b,c // a,c by A4,Th7;
    hence thesis by A6,A5,Def1;
  end;
  hence thesis by A3;
end;
