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 Th12:
  not a,b // a,c & p<>q implies not p,q // p,a or not p,q // p,b
  or not p,q // p,c
proof
  assume that
A1: not a,b // a,c and
A2: p<>q;
  now
    assume that
A3: a<>p and
A4: p,q // p,a and
A5: p,q // p,b and
A6: p,q // p,c;
    p,a // p,c by A2,A4,A6,Def1;
    then
A7: a,p // a,c by Def1;
    p,a // p,b by A2,A4,A5,Def1;
    then a,p // a,b by Def1;
    hence contradiction by A1,A3,A7,Def1;
  end;
  hence thesis by A1,A2,Def1;
end;
