reserve AS for AffinSpace;
reserve a,a9,b,b9,c,d,o,p,q,r,s,x,y,z,t,u,w for Element of AS;

theorem Th8:
  x<>y & LIN x,y,z & x,y // z,t implies LIN x,y,t
proof
  assume that
A1: x<>y and
A2: LIN x,y,z and
A3: x,y // z,t;
  now
    x,y // x,z by A2;
    then x,z // z,t by A1,A3,Th4;
    then z,x // z,t by Th3;
    then LIN z,x,t;
    then
A4: LIN x,z,t by Th5;
    assume
A5: z<>x;
A6: LIN x,z,x by Th6;
    LIN x,z,y by A2,Th5;
    hence thesis by A5,A4,A6,Th7;
  end;
  hence thesis by A3;
end;
