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;

theorem Th1:
  ({a,b} on P implies {b,a} on P) & ({a,b,c} on P implies {a,c,b}
  on P & {b,a,c} on P & {b,c,a} on P & {c,a,b} on P & {c,b,a} on P) & (a on P,Q
implies a on Q,P) & (a on P,Q,R implies a on P,R,Q & a on Q,P,R & a on Q,R,P &
  a on R,P,Q & a on R,Q,P)
proof
  thus {a,b} on P implies {b,a} on P;
  thus {a,b,c} on P implies {a,c,b} on P & {b,a,c} on P & {b,c,a} on P & {c,a,
  b} on P & {c,b,a} on P
  proof
    assume
A1: {a,b,c} on P;
    then
A2: c on P by INCSP_1:2;
    a on P & b on P by A1,INCSP_1:2;
    hence thesis by A2,INCSP_1:2;
  end;
  thus a on P,Q implies a on Q,P;
  assume
A3: a on P,Q,R;
  then
A4: a on R;
  a on P & a on Q by A3;
  hence thesis by A4;
end;
