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 Th10:
  u<>z & LIN x,y,u & LIN x,y,z & LIN u,z,w implies LIN x,y,w
proof
  assume that
A1: u<>z and
A2: LIN x,y,u and
A3: LIN x,y,z and
A4: LIN u,z,w;
  now
    assume
A5: x<>y;
    LIN x,y,x by Th6;
    then
A6: LIN z,u,x by A2,A3,A5,Th7;
    LIN x,y,y by Th6;
    then
A7: LIN z,u,y by A2,A3,A5,Th7;
    LIN z,u,w by A4,Th5;
    hence thesis by A1,A7,A6,Th7;
  end;
  hence thesis by Th6;
end;
