reserve F for Field;
reserve a,b,c,d,p,q,r for Element of MPS(F);
reserve e,f,g,h,i,j,k,l,m,n,o,w for Element of [:the carrier of F,the carrier
  of F,the carrier of F:];
reserve K,L,M,N,R,S for Element of F;
reserve FdSp for FanodesSp;
reserve a,b,c,d,p,q,r,s,o,x,y for Element of FdSp;

theorem
  not o,a,b are_collinear implies not o,a,x are_collinear or not o,b,x
  are_collinear or o=x
proof
  assume
A1: not o,a,b are_collinear;
  now
    assume that
A2: o,a,x are_collinear and
A3: o,b,x are_collinear;
    a,o,x are_collinear by A2,Th10;
    then
A4: a,o '||' a,x;
    b,o,x are_collinear by A3,Th10;
    then
A5: b,o '||' b,x;
    not a,b,o are_collinear by A1,Th10;
    then not a,b '||' a,o;
    hence thesis by A4,A5,PARSP_1:33;
  end;
  hence thesis;
end;
