reserve IPP for IncProjSp;
reserve a,b,c,d,p,q,o,r,s,t,u,v,w,x,y for POINT of IPP;
reserve A,B,C,D,L,P,Q,S for LINE of IPP;

theorem Th13:
  (ex A,a,b,c,d st a on A & b on A & c on A & d on A & a,b,c,d
are_mutually_distinct) implies for B ex p,q,r,s st p on B & q on B & r on B &
  s on B & p,q,r,s are_mutually_distinct
proof
  given A,a,b,c,d such that
A1: a on A & b on A & c on A & d on A & a,b,c,d are_mutually_distinct;
  let B;
  consider x such that
A2: x on A by Th7;
  consider y such that
A3: y on B by Th7;
  consider C such that
A4: x on C and
A5: y on C by INCPROJ:def 5;
  ex p1,q1,r1,s1 being POINT of IPP st p1 on C & q1 on C & r1 on C & s1
  on C & p1,q1,r1,s1 are_mutually_distinct by A1,A2,A4,Lm1;
  hence thesis by A3,A5,Lm1;
end;
