theorem Th20:
  a,b,c,d is_a_quadrangle & {a,b} on P & {a,c} on Q & {a,d} on R
  implies P,Q,R are_mutually_distinct
proof
  assume that
A1: a,b,c,d is_a_quadrangle and
A2: {a,b} on P and
A3: {a,c} on Q and
A4: {a,d} on R;
  c,d,a is_a_triangle by A1;
  then a,c,d is_a_triangle by Th11;
  then
A5: Q<>R by A3,A4,Th19;
  d,a,b is_a_triangle by A1;
  then a,d,b is_a_triangle by Th11;
  then
A6: R<>P by A2,A4,Th19;
  a,b,c is_a_triangle by A1;
  then P<>Q by A2,A3,Th19;
  hence thesis by A5,A6,ZFMISC_1:def 5;
end;
