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 Th19:
  not a,b // a,c & a,b // p,q & a,c // p,r1 & a,c // p,r2 & b,c //
  q,r1 & b,c // q,r2 implies r1=r2
proof
  assume that
A1: not a,b // a,c and
A2: a,b // p,q and
A3: a,c // p,r1 and
A4: a,c // p,r2 and
A5: b,c // q,r1 and
A6: b,c // q,r2;
A7: now
    b<>c by A1,Th1;
    then
A8: q,r1 // q,r2 by A5,A6,Def1;
    a<>c by A1,Def1;
    then
A9: p,r1 // p,r2 by A3,A4,Def1;
    assume p<>q;
    then not p,q // p,r1 by A1,A2,A3,A5,Th16;
    hence thesis by A9,A8,Th18;
  end;
  now
    assume
A10: p=q;
    then p=r1 by A1,A3,A5,Th17;
    hence thesis by A1,A4,A6,A10,Th17;
  end;
  hence thesis by A7;
end;
