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 Th7:
  x<>y & LIN x,y,z & LIN x,y,t & LIN x,y,u implies LIN z,t,u
proof
  assume that
A1: x<>y and
A2: LIN x,y,z and
A3: LIN x,y,t and
A4: LIN x,y,u;
A5: now
A6: x,y // x,z by A2;
    x,y // x,u by A4;
    then x,z // x,u by A1,A6,Th4;
    then
A7: z,x // z,u by DIRAF:40;
    x,y // x,t by A3;
    then x,z // x,t by A1,A6,Th4;
    then
A8: z,x // z,t by DIRAF:40;
    assume x<>z;
    then z,t // z,u by A8,A7,Th4;
    hence thesis;
  end;
  x=z implies thesis by A1,A3,A4,Th4;
  hence thesis by A5;
end;
