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 Th14:
  not a,b // a,c implies not a,b // c,a & not b,a // a,c & not b,a
  // c,a & not a,c // a,b & not a,c // b,a & not c,a // a,b & not c,a // b,a &
not b,a // b,c & not b,a // c,b & not a,b // b,c & not a,b // c,b & not b,c //
b,a & not b,c // a,b & not c,b // a,b & not c,b // b,a & not c,b // c,a & not c
,b // a,c & not b,c // c,a & not b,c // a,c & not c,a // c,b & not c,a // b,c &
  not a,c // b,c & not a,c // c,b
proof
  assume
A1: not a,b // a,c;
A2: now
    assume a,c // c,b;
    then c,a // c,b by Th6;
    hence contradiction by A1,Th7;
  end;
  assume
A3: not thesis;
  not b,a // b,c by A1,Th7;
  hence thesis by A1,A3,A2,Th6;
end;
