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
  a,b // a,x & b,c // b,x & c,a // c,x implies a,b // a,c
proof
  assume that
A1: a,b // a,x and
A2: b,c // b,x and
A3: c,a // c,x;
  now
    assume
A4: a<>x;
    a,x // a,b & a,x // a,c by A1,A3,Th7;
    hence thesis by A4,Def1;
  end;
  hence thesis by A2,Th7;
end;
