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;

theorem
  A<>B & a on A & q|'A & a<>A*B implies (q*a)*B on B & (q*a)*B|'A
proof
  assume that
A1: A<>B and
A2: a on A and
A3: q|'A and
A4: a<>A*B;
  set D=q*a;
A5: a on D & q on D by A2,A3,Th16;
  set d=D*B;
A6: G is configuration by Th23;
A7: a|'B by A1,A2,A4,Th24;
  then
A8: q*a<>B by A2,A3,Th16;
  hence (q*a)*B on B by Th24;
  assume
A9: d on A;
  d on D by A8,Th24;
  then a = d by A2,A3,A6,A9,A5;
  hence thesis by A7,A8,Th24;
end;
