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,m,m9 being POINT of G, I being LINE
  of G st m on I & m9 on I & m<>m9 & e|'I holds m*e<>m9*e & e*m<>e*m9
proof
  let G be IncProjectivePlane, e,m,m9 be POINT of G, I be LINE of G such that
A1: m on I and
A2: m9 on I and
A3: m<>m9 and
A4: e |'I;
  set L1=m*e,L2=m9*e;
A5: now
    m on L1 by A1,A4,Th16;
    then
A6: m on I,L1 by A1;
    e on L1 by A1,A4,Th16;
    then
A7: m=I*L1 by A4,A6,Def9;
    assume
A8: m*e=m9*e;
    m9 on L2 by A2,A4,Th16;
    then
A9: m9 on I,L2 by A2;
    e on L2 by A2,A4,Th16;
    hence contradiction by A3,A4,A8,A9,A7,Def9;
  end;
  now
    assume
A10: e*m=e*m9;
    m*e=e*m by A1,A4,Th16;
    hence contradiction by A2,A4,A5,A10,Th16;
  end;
  hence thesis by A5;
end;
