reserve x,y for set;
reserve X for non empty set;
reserve a,b,c,d for Element of X;
reserve S for OAffinSpace;
reserve a,b,c,d,p,q,r,x,y,z,t,u,w for Element of S;

theorem
  u<>z & x,y,u are_collinear & x,y,z are_collinear & u,z,w are_collinear
  implies x,y,w are_collinear
proof
  assume that
A1: u<>z and
A2: x,y,u are_collinear & x,y,z are_collinear and
A3: u,z,w are_collinear;
  now
    assume
A4: x<>y;
    x,y,x are_collinear by Th31;
    then
A5: z,u,x are_collinear by A2,A4,Th32;
    x,y,y are_collinear by Th31;
    then
A6: z,u,y are_collinear by A2,A4,Th32;
    z,u,w are_collinear by A3,Th30;
    hence thesis by A1,A6,A5,Th32;
  end;
  hence thesis by Th31;
end;
