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 Th16:
  not a,b // a,c & a,b // p,q & a,c // p,r & b,c // q,r & p<>q
  implies not p,q // p,r
proof
  assume that
A1: not a,b // a,c and
A2: a,b // p,q and
A3: a,c // p,r and
A4: b,c // q,r and
A5: p<>q;
  now
    assume p=r;
    then
A6: p,q // b,c by A4,Th6;
    p,q // a,b by A2,Th6;
    then a,b // b,c by A5,A6,Def1;
    then b,a // b,c by Th4;
    hence contradiction by A1,Th7;
  end;
  hence thesis by A1,A2,A3,A5,Th15;
end;
