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 Th13:
  a<>b & a,b,p are_collinear & a,b,q are_collinear & a,b,r
  are_collinear implies p,q,r are_collinear
proof
  assume that
A1: a<>b and
A2: a,b,p are_collinear and
A3: a,b,q are_collinear and
A4: a,b,r are_collinear;
A5: a,b '||' a,p by A2;
  a,b '||' a,r by A4;
  then
A6: a,b '||' p,r by A5,PARSP_1:35;
  a,b '||' a,q by A3;
  then a,b '||' p,q by A5,PARSP_1:35;
  then p,q '||' p,r by A1,A6,PARSP_1:def 12;
  hence thesis;
end;
