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
  a<>b & a,b,c are_collinear & a,b '||' c,d implies a,c '||' b,d
proof
  assume that
A1: a<>b and
A2: a,b,c are_collinear and
A3: a,b '||' c,d;
  now
A4: a,b '||' a,c by A2;
    then a,b '||' c,b by PARSP_1:24;
    then c,b '||' c,d by A1,A3,PARSP_1:def 12;
    then
A5: b,c '||' b,d by PARSP_1:24;
    assume
A6: b<>c;
    b,c '||' a,c by A4,PARSP_1:24;
    hence thesis by A6,A5,PARSP_1:def 12;
  end;
  hence thesis by A3;
end;
