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 Th21:
  G is configuration & a on P,Q,R & P,Q,R are_mutually_distinct &
  a|'A & p on A,P & q on A,Q & r on A,R implies p,q,r are_mutually_distinct
proof
  assume that
A1: G is configuration and
A2: a on P,Q,R and
A3: P,Q,R are_mutually_distinct and
A4: a|'A and
A5: p on A,P and
A6: q on A,Q and
A7: r on A,R;
A8: a on R & r on R by A2,A7;
A9: a on Q & q on Q by A2,A6;
  Q<>R & q on A by A3,A6,ZFMISC_1:def 5;
  then
A10: q<>r by A1,A4,A9,A8;
A11: p on P by A5;
A12: a on P & p on A by A2,A5;
  R<>P by A3,ZFMISC_1:def 5;
  then
A13: r<>p by A1,A4,A12,A11,A8;
  P<>Q by A3,ZFMISC_1:def 5;
  then p<>q by A1,A4,A12,A11,A9;
  hence thesis by A10,A13,ZFMISC_1:def 5;
end;
