reserve G for IncProjStr;
reserve a,a1,a2,b,b1,b2,c,d,p,q,r for POINT of G;
reserve A,B,C,D,M,N,P,Q,R for LINE of G;
reserve G for IncProjectivePlane;
reserve a,q for POINT of G;
reserve A,B for LINE of G;
reserve G for IncProjSp;
reserve a,b,c,d for POINT of G;
reserve P for LINE of G;
reserve G for IncProjectivePlane;

theorem
  for G being IncProjectivePlane, e being POINT of G, I,L1,L2 being LINE
  of G st e on L1 & e on L2 & L1<>L2 & e|'I holds I*L1<>I*L2 & L1*I<>L2*I
proof
  let G be IncProjectivePlane, e be POINT of G, I,L1,L2 be LINE of G such that
A1: e on L1 and
A2: e on L2 and
A3: L1<>L2 and
A4: e |'I;
  set p1=I*L1,p2=I*L2;
A5: now
    p1 on L1 by A1,A4,Th24;
    then
A6: {e,p1} on L1 by A1,INCSP_1:1;
    p1 on I by A1,A4,Th24;
    then
A7: L1=e*p1 by A4,A6,Def8;
    assume
A8: I*L1=I*L2;
    p2 on L2 by A2,A4,Th24;
    then
A9: {e,p2} on L2 by A2,INCSP_1:1;
    p2 on I by A2,A4,Th24;
    hence contradiction by A3,A4,A8,A9,A7,Def8;
  end;
  now
    assume
A10: L1*I=L2*I;
    L1*I=I*L1 by A1,A4,Th24;
    hence contradiction by A2,A4,A5,A10,Th24;
  end;
  hence thesis by A5;
end;
