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
  o<>a & o<>b & o,a,b are_collinear & o,a,p are_collinear & o,b,q
  are_collinear implies a,b '||' p,q
proof
  assume that
A1: o<>a and
A2: o<>b and
A3: o,a,b are_collinear and
A4: o,a,p are_collinear and
A5: o,b,q are_collinear;
A6: now
A7: o,a '||' o,b by A3;
    o,a '||' o,p by A4;
    then
A8: o,b '||' o,p by A1,A7,PARSP_1:def 12;
    o,b '||' o,q by A5;
    then o,p '||' o,q by A2,A8,PARSP_1:def 12;
    then
A9: o,p '||' p,q by PARSP_1:24;
    o,b '||' a,b by A7,PARSP_1:24;
    then
A10: a,b '||' o,p by A2,A8,PARSP_1:def 12;
    assume o<>p;
    hence thesis by A10,A9,PARSP_1:26;
  end;
  now
    assume
A11: o=p;
    then p,a '||' p,b by A3;
    then
A12: a,b '||' p,b by PARSP_1:24;
    p,b '||' p,q by A5,A11;
    hence thesis by A2,A11,A12,PARSP_1:26;
  end;
  hence thesis by A6;
end;
