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 a,b '||' c,d & a,b,p are_collinear & a,b,q are_collinear & c,d,p
  are_collinear & c,d,q are_collinear implies p=q
proof
  assume that
A1: not a,b '||' c,d and
A2: a,b,p are_collinear & a,b,q are_collinear and
A3: c,d,p are_collinear & c,d,q are_collinear;
  c,d '||' p,q by A3,Th15;
  then
A4: p,q '||' c,d by PARSP_1:23;
  a,b '||' p,q by A2,Th15;
  then p,q '||' a,b by PARSP_1:23;
  hence thesis by A1,A4,PARSP_1:def 12;
end;
