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 Th16:
  not a,b,c are_collinear & a,b '||' c,d implies not a,b,d are_collinear
proof
  assume that
A1: not a,b,c are_collinear and
A2: a,b '||' c,d;
A3: now
    assume that
A4: c <>d and
A5: a<>d;
    a,c '||' c,a & c <>a by A1,PARSP_1:25;
    then not c,d,a are_collinear by A1,A2,A4,Th11;
    then
A6: not d,c,a are_collinear by Th10;
A7: d,a '||' a,d by PARSP_1:25;
    a<>b & d,c '||' a,b by A1,A2,Th12,PARSP_1:23;
    hence thesis by A5,A6,A7,Th11;
  end;
  a <> d by A1,A2,PARSP_1:23;
  hence thesis by A1,A3;
end;
